Lurbinectedin With Berzosertib, an ATR Kinase Inhibitor in Small Cell Cancers and High-Grade Neuroendocrine Cancers

* INCLUSION CRITERIA:
* Both Phase I and Phase II:
* \>= 18 years of age.
* ECOG performance status \<= 2
* Measurable disease, per RECIST 1.1. Individuals with evaluable, but not measurable disease will be eligible for Phase I.
* Adequate organ functions

  * Hemoglobin \>= 9.0 g/dL
  * Absolute neutrophil count \>= 1.5x10\^9/L
  * Platelets \>= 100x10\^9/L
  * Total Bilirubin \<= 2.0 mg/dL
  * Transaminases \<= 2 x ULN or if liver metastases were present, \<= 3 x ULN
  * Creatinine \<= 1.5 mg/dL or creatinine clearance by Cockcroft-Gault formula \>= 60 mL/min
* Ability to understand and the willingness to sign a written informed consent document.
* Individuals of child-bearing potential (IOCBP) and individuals able to father a child must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, during study participation and for 6 months after the last dose of berzosertib/lurbinectedin for IOCBP and for 4 months after lurbinectedin or 3 months after berzosertib for individuals able to father children.

Phase I:

* Histologically confirmed advanced solid cancers will be eligible.
* At least one prior chemotherapy

Phase II:

\- Histological confirmation of SCLC or HGNEC. Although NCI confirmation of pathology is not required prior to starting treatment, every effort will be made to obtain outside pathology to be reviewed by an NCI pathologist.

EXCLUSION CRITERIA:

* Individuals with tumor amenable to potentially curative therapy.
* Currently receiving any other investigational agents.
* Received chemotherapy, or undergone major surgery within the prior 2 weeks and radiotherapy within the last 24 hours.
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to (study agent) or other agents used in study.
* Symptomatic brain metastases will be excluded from trial secondary to poor prognosis. However, individuals who have had treatment for their brain metastasis and whose brain disease is stable without steroid therapy for 1 week or on physiologic doses of steroids may be enrolled.
* Requirement for any medications or substances that are strong inhibitors or inducers of CYP3A during the course of the study are ineligible.
* Evidence of severe or uncontrolled systemic disease, or any concurrent condition, which could compromise participation in the study, including, but not limited to, active or uncontrolled infection, immune deficiencies, Hepatitis B, Hepatitis C, uncontrolled diabetes, uncontrolled hypertension, symptomatic congestive heart failure, unstable angina pectoris, myocardial infarction within the past 6 months, uncontrolled cardiac arrhythmia, stroke/cerebrovascular accident within the past 6 months, or psychiatric illness/social situations which would jeopardize compliance with the protocol.
* HIV-positive on or off combination antiretroviral therapy are ineligible.
* Pregnant individuals are excluded from this study.