Testing the Sequential Combination of the Anti-cancer Drugs Olaparib Followed by Adavosertib (AZD1775) in Patients With Advanced Solid Tumors With Selected Mutations and PARP Resistance, STAR Study

Inclusion Criteria:

* Subjects must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative measures do not exist or are no longer effective
* Patients in dose expansion Cohort A (intrinsic resistance), must have:

  * Prior treatment with PARP inhibitors
  * Disease progression per Response Evaluation Criteria in Solid Tumors (RECIST) at 1st re-staging, and
  * Germline or somatic mutations in BRCA1 or BRCA2
* Patients in dose expansion Cohort B (acquired resistance) must have:

  * Prior treatment with PARP inhibitors,
  * Complete/partial response followed by disease progression per RECIST, and
  * Germline or somatic mutations in any of the following deoxyribonucleic acid (DNA) damage response (DDR) genes: BRCA1, BRCA2, BRIP1, FANCA, PALB2, or the non-DDR gene marker cyclin E amplification. Local testing in Clinical Laboratory Improvement Act (CLIA)-certified laboratory will be accepted. All alterations will be reviewed by MD Anderson's Precision Oncology Decision Support (PODS) team. No variants of uncertain significance (VUS) will be allowed as the qualifying genetic mutation. Recruitment of patients with relevant molecular aberrations in the dose escalation phase is encouraged but not mandated.
* Subjects must have RECIST measurable disease and a tumor that is safely accessible for biopsy and must be willing to undergo biopsy
* Subjects must have received at least one line of systemic therapy in the advanced/metastatic setting. Subjects with diseases without known effective options, and subject who have declined standard or care therapy prior to study introduction are also eligible
* Any prior palliative radiation therapy must have been completed at least 14 days prior to the start of study drugs, and patients must have recovered from any acute adverse effects prior to the start of study treatment
* Human immunodeficiency virus (HIV)-infected (HIV1/2 antibody-positive) patients may participate if they meet all the following eligibility requirements:

  * They must be on an anti-retroviral regimen with evidence of at least two undetectable viral loads within the past 6 months on this same regimen; the most recent undetectable viral load must be within the past 12 weeks
  * They must have a CD4 count \>= 250 cells/uL over the past 6 months on this same anti-retroviral regimen and must not have had a CD4 count \< 200 cells/ul over the past 2 years, unless it was deemed related to the cancer and/or chemotherapy-induced bone marrow suppression. For patients who have received chemotherapy in the past 6 months, a CD4 count \< 250 cells/ul during chemotherapy is permitted as long as viral loads were undetectable during this same chemotherapy
  * They must have an undetectable viral load and a CD4 count \>= 250 cells/uL within 7 days of enrollment
  * They must not be currently receiving prophylactic therapy for an opportunistic infection and must not have had an opportunistic infection within the past 6 months. Monitoring for HIV-infected patients should include:

    * Viral load and CD4 count every 12 weeks (q12w)
    * If CD4 count drops to less than 200 cells/ul while on study, initiate viral load test. If viral load proves undetectable at this time, continue CD4 and viral load checks every 8 weeks (q8w). If 2 consecutive viral load tests are undetectable, revert to q12w testing for CD4 and viral load testing
    * If an opportunistic infection occurs with a CD4 count of \< 200 cells/ul, hold study treatment. Initiate treatment of the infection and continue to hold study treatment; once clinically stable, CD4 count is \> 200 cells/ul and viral load has remained undetectable, reinitiate study treatment
* Patients with a past or resolved hepatitis B virus (HBV) infection (defined as the presence of hepatitis B core antibody and absence of HBsAg) are eligible. Active HBV is defined by a known positive HBV surface antigen (HBsAg) result. Patients positive for hepatitis C virus (HBC) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV RNA. Patients with known active HBV or HBC infection are ineligible
* Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression within 28 days
* Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy
* Absolute neutrophil count (ANC) \>= 1500/uL (within 7 days of study drugs initiation)
* Hemoglobin (Hgb) \>= 10 g/dL (within 7 days of study drugs initiation) with no blood transfusion in the past 28 days
* Platelets \>= 100,000/uL (within 7 days of study drugs initiation)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvate transaminase \[SGPT\]) =\< 2.5 X upper limit of normal (ULN) (within 7 days of study drugs initiation), unless liver metastases are present in which case they must be =\< 5 X ULN
* Serum bilirubin within normal limits (WNL) (within 7 days of study drugs initiation) or =\< 1.5 X ULN in patients with liver metastases, or total bilirubin =\< 3 X ULN with direct bilirubin WNL in patients with well-documented Gilbert's syndrome
* Serum creatinine =\< 1.5 X ULN (within 7 days of study drugs initiation). If elevated, check creatinine clearance (CrCl) with cut off \>= 51 mL/min as calculated by the Cockcroft-Gault method or based on a 24-hour urine test (confirmation of creatinine clearance is only required when serum creatinine is \> 1.5 X institutional ULN)
* Estimated CrCl (glomerular filtration rate \[GFR\]) = (140-age \[years\]) x (weight \[kg\]) x Fa (72 x serum creatinine mg/dL) (a where F = 0.85 for females and F = 1 for males)
* The effects of AZD1775 and olaparib on the developing human fetus are either unclear or are known to be teratogenic, women of childbearing potential and their partners, who are sexually active, must agree to the use of TWO highly effective forms of contraception in combination. This should be started from the signing of the informed consent and continue throughout the period of taking study treatment and for at least 1 month after last dose of study drug(s), or they must totally/truly abstain from any form of sexual intercourse. Male patients must use a condom during treatment and for 3 months after the last dose of study drug when having sexual intercourse with a pregnant woman or with a woman of childbearing potential. Female partners of male patients should also use a highly effective form of contraception if they are of childbearing potential. Male patients should not donate sperm throughout the period of taking study drugs and for 3 months following the last dose of study drugs.

  * Acceptable non-hormonal birth control methods include:

    * Total/True abstinence: When the patient refrains from any form of sexual intercourse and this is in line with their usual and/or preferred lifestyle; this must continue for the total duration of the trial and for at least 1 month after the last dose of study drug for women of child bearing potential. For male patients, 3 months after last dose. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods, or declaration of abstinence solely for the duration of a trial\] and withdrawal are not acceptable methods of contraception.)
    * Vasectomized sexual partner PLUS male condom. With participant assurance that partner received post-vasectomy confirmation of azoospermia
    * Tubal occlusion PLUS male condom
    * Intrauterine device (IUD) PLUS male condom. Provided coils are copper-banded
  * Acceptable hormonal methods:

    * Normal and low dose combined oral pills PLUS male condom
    * Cerazette (desogestrel) PLUS male condom. Cerazette is currently the only highly efficacious progesterone based pill
    * Hormonal shot or injection (e.g., Depo-Provera) PLUS male condom
    * Etonogestrel implants (e.g., Implanon, Norplant) PLUS male condom
    * Norelgestromin / ethinyl estradiol (EE) transdermal system PLUS male condom
    * Intrauterine system (IUS) device (e.g., levonorgestrel releasing IUD-Mirena) PLUS male condom
    * Intravaginal device (e.g., EE and etonogestrel) PLUS male condom
    * In the case of use of oral contraception, women should be stable on the same pill before taking study treatment
    * Note: oral contraceptives are allowed but should be used in conjunction with a barrier method of contraception due to unknown effect of drug-drug interaction. Women are considered post-menopausal and not of childbearing potential if they have had 12 or more months of natural (spontaneous) amenorrhea following cessation of exogenous hormonal treatment with an appropriate clinical profile (e.g. age appropriate, history of vasomotor symptoms) or have had surgical bilateral oophorectomy (with or without hysterectomy). In the case of oophorectomy alone, she will be considered not of childbearing potential only when her reproductive status has been confirmed by follow up hormone level assessment
* Male patients will be advised to arrange for the freezing of sperm samples prior to the start of the study should they wish to father children while on study drugs or during the 3 months after stopping study drugs
* Male or female patient \>= 18 years of age. Because no dosing or adverse event data are currently available on the use of AZD1775 in combination with olaparib in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) score of 0-1 (Karnofsky \>= 70%)
* Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible. Has read and understands the informed consent form (ICF) and has given written informed consent (IC) prior to any study procedures
* Willingness and ability to comply with study and follow-up procedures

Exclusion Criteria:

* History of allergic reactions attributed to compounds of similar chemical or biologic composition to AZD1775 or olaparib
* Use of anti-cancer treatment drug =\< 28 days or 5 half-lives (whichever is shorter) prior to the first dose of study treatment. For drugs for which 5 half-lives is =\< 21 days, a minimum of 10 days between termination of the prior treatment and administration of study treatment is required
* Use of radiotherapy (except for palliative reasons) within =\< 28 days prior to study treatment
* No other anti-cancer therapy (chemotherapy, immunotherapy, hormonal anticancer therapy radiotherapy), biological therapy or other novel agent is to be permitted while the patient is receiving study medication. Patients with castration-resistant prostate cancer on luteinizing hormone-releasing hormone (LHRH) analogue treatment for more than 6 months are allowed entry into the study and may continue at the discretion of the investigator
* Concomitant use of CYP3A inducers/inhibitors:

  * Concomitant use of known strong CYP3A inhibitors (e.g., itraconazole, telithromycin, clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir, saquinavir, nelfinavir, boceprevir, telaprevir) or moderate CYP3A inhibitors (e.g., ciprofloxacin, erythromycin, diltiazem, fluconazole, verapamil). The required washout period prior to starting olaparib is 2 weeks
  * Concomitant use of known strong (e.g., phenobarbital, enzalutamide, phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine and St John's Wort) or moderate CYP3A inducers (e.g., bosentan, efavirenz, modafinil). The required washout period prior to starting study treatment is 5 weeks for enzalutamide or phenobarbital and 3 weeks for other agents
* Major surgical procedures =\< 28 days of beginning study treatment, or minor surgical procedures =\< 7 days. Patients must have recovered from any of the effects of any major surgery. No waiting period required following port-a-cath placement or other central venous access placement
* Known malignant central nervous system (CNS) disease other than neurologically stable, treated brain metastases - defined as metastasis having no evidence of progression or hemorrhage for at least 2 weeks after treatment (including brain radiotherapy). Must be off any systemic corticosteroids for the treatment of brain metastases for at least 14 days prior to enrollment. Subjects with brain metastases must have completed treatment, either surgery or radiation, and be stable for at least 28 days off steroid prior to screening. A brain magnetic resonance imaging (MRI) demonstrating there is no current evidence or progressive brain metastases is required in subjects with previous brain metastasis. Patients with breast tissue expanders may have brain computed tomography (CT) for assessment
* Patients with either previous or current myelodysplastic syndrome (MDS)/acute myeloid leukemia (AML) or features suggestive of MDS/AML (e.g., persistent anemia or other blood dyscrasias) are excluded because olaparib and AZD1775 are agents with the potential to induce MDS/AML
* AZD1775 should not be given to patients who have a history of Torsades de pointes (TdP) unless all risk factors that contributed to TdP have been corrected
* Any of the following cardiac diseases currently or within the last 6 months:

  * Unstable angina pectoris
  * Acute myocardial infarction
  * Congestive heart failure \>= class 2 (as defined by New York Heart Association \[NYHA\])
  * Conduction abnormality not controlled with pacemaker or medication
  * Significant ventricular or supraventricular arrhythmias (patients with chronic rate-controlled atrial fibrillation in the absence of other cardiac abnormalities are eligible)
* Participants with a mean resting corrected QT interval (QTc) \>= 480 msec at study entry, as calculated by the Frederica formula (QTcF) by institutional standards obtained from an electrocardiogram (ECG) or congenital long QT syndrome. (Note: if one ECG demonstrates a QTcF \> 480 msec, then a mean QTcF of =\< 480 msec obtained from 3 ECGs 2-5 minutes apart is required at study entry
* Patients considered a poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 3 months) myocardial infarction, uncontrolled major seizure disorder, unstable spinal cord compression, superior vena cava syndrome, extensive interstitial bilateral lung disease on high resolution computed tomography (HRCT) scan or any psychiatric disorder that prohibits obtaining informed consent
* Patients with uncontrolled intercurrent illness
* Patients with psychiatric illness/social situations, or other psychological, familial, sociological, or geographical conditions that would limit compliance with study requirements
* Persistent grade \> 1 toxicity from prior cancer therapy (except alopecia)
* Patients with previous allogeneic, bone marrow, or double umbilical cord blood transplants are not allowed
* Consumption of grapefruit juice while on study drugs
* Involvement in the planning and/or conduct of the study
* Patients unable to swallow orally administered medications and patients with gastrointestinal disorders likely to interfere with absorption of the study medication
* Pregnant or breastfeeding women are excluded from this study because AZD1775 and olaparib are agents with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with AZD1775 and olaparib, breastfeeding should be discontinued if the mother is treated with AZD1775 and olaparib
* Other malignancy unless curatively treated with no evidence of disease for \>= 5 years except: adequately treated non-melanoma skin cancer, curatively treated in situ cancer of the cervix, ductal carcinoma in situ (DCIS), stage 1, grade 1 endometrial carcinoma