Inclusion Criteria:
1. Participants must have histologically or cytologically confirmed advanced solid tumors (any solid tumor type) with:
Phase I, Dose Escalation: Germline and/or somatic mutation\* in one or more of the following genes: BRCA1, BRCA2, PALB2, ATM, and/or CHEK2.
Phase Ib, Dose Expansion\*\*:
1. Expansion Cohort A (n=6): Germline mutation\* (with or without accompanying somatic mutation) in one or more of the following genes: BRCA1, BRCA2, PALB2, ATM, and/or CHEK2;
2. Expansion Cohort B (n=6): Germline and/or somatic mutation\* in one or more of the following genes: BRCA1, BRCA2, PALB2, ATM, and/or CHEK2.
* Testing for DNA repair mutations should occur prior to study consent or enrollment via a CLIA-approved test.
2. Has measurable disease per RECIST 1.1 as assessed by the investigator. Lesions situated in previously irradiated areas are considered measurable if progression has been demonstrated in such lesions.
3. Participants may have received any lines of prior therapy and is refractory or intolerant to therapy approved for their condition or unwilling to receive currently approved therapy.
4. Prior PARP inhibitors are allowed, provided the following two criteria are met:
1. Participant has NOT required toxicity related dose reductions or dose delays during prior PARP inhibitor treatment; and
2. Participant has NOT experienced any allergic reaction to PARP inhibitors.
5. Age \>=18 years
6. Eastern Cooperative Oncology Group (ECOG) performance status \<2, or Karnofsky \>60%
7. Demonstrates adequate organ function as defined below:
Adequate bone marrow function:
1. hemoglobin \>=10.0 g/dl
2. absolute neutrophil count \>=1,500/microliter (mcL)
3. platelets \>=100,000/mcL
Adequate hepatic function:
1. total bilirubin ≤ 1.5 x institutional upper limit normal (ULN)
2. aspartate aminotransferase (AST)/(SGOT) \<= 2.5 x institutional ULN
3. alanine aminotransferase (ALT)/(SGPT) \<= 2.5 x institutional ULN
4. creatinine \<= 1.5 x institutional ULN or creatinine clearance Glomerular filtration rate (GFR) \>= 50 mL/min/1.73 m\^2, calculated using the Cockcroft-Gault equation.
8. Ability to understand and the willingness to sign a written informed consent document.
9. Human immunodeficiency virus (HIV)-infected individuals on effective antiretroviral therapy with undetectable viral load within 6 months are eligible for this trial.
10. For participants with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated.
11. Individuals with a history of hepatitis C virus (HCV) infection must have been treated and cured. Individuals with HCV infection who are currently on treatment could be eligible if HCV viral load is undetectable.
12. Individuals with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression for at least 4 weeks. Participants need to be without requirement for steroid treatment for at least 14 days prior to the first dose of study intervention. A participant with one or two lesions that have been definitely treated with resection or focal radiation and has no symptoms is eligible after 2 weeks.
13. Based on findings from human data and/or animal studies, and their mechanisms of action, ASTX727 and olaparib can cause fetal harm when administered to a pregnant woman. For this reason, females of child-bearing potential (defined below) must agree to use adequate contraception including hormonal or barrier methods or strict abstinence for the duration of study treatment and for 6 months after last administration of study treatment. Males (with female partners of reproductive potential or who are pregnant) treated or enrolled on this protocol also must agree to use adequate contraception for the duration of study treatment, and for 3 months after last administration of study treatment. Should an individual participating in this study (or the partner of an individual participating in the study) become pregnant or suspect pregnancy, they should inform the treating physician immediately. A female is considered to be of childbearing potential (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice), unless it is documented that the individual meets either of the following two criteria: (1) has reached a postmenopausal state (\>= 12 continuous months of amenorrhea with no identified cause other than menopause); or (2) has undergone surgical sterilization (i.e., hysterectomy and/or bilateral oophorectomy for removal of uterus and/or ovaries).
14. Individuals with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial.
Note: Diagnosis of myelodysplastic syndrome (MDS) or acute myeloid leukemia (AML) are excluded per Exclusion # 7.
Exclusion Criteria:
1. Has received systemic anticancer therapies within 3 weeks of first dose, radiation within 2 weeks, antibody therapy within 4 weeks. Concomitant administration of Luteinizing hormone-releasing hormone (LHRH) analogues for prostate cancer and somatostatin analogues for neuroendocrine tumors are allowed as per standard of care.
2. Has not recovered from adverse events due to prior anti-cancer therapy to \<= grade 1 (CTCAE v5.0) or baseline (other than alopecia).
3. Receipt of any other investigational agents or devices within 3 weeks prior to initiation of trial therapy.
4. Unable to swallow oral medications
5. Individuals who are breast-feeding/chest-feeding (because of the potential for serious adverse reactions in breastfeed infants from olaparib and ASTX727). Study participants who are lactating must agree to discontinue breast-feeding/chest-feeding for the duration of study treatment and for 1 month after the final dose of trial therapy.
6. Individuals who are pregnant (because of the potential for olaparib and ASTX727 to cause serious adverse reactions to the unborn child). Females of childbearing potential (defined below) must have a negative urine or serum pregnancy test within 72 hours prior to first administration of study drug. A female is considered to be of childbearing potential (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice), unless it is documented that the individual meets either of the following two criteria: (1) has reached a postmenopausal state (\>= 12 continuous months of amenorrhea with no identified cause other than menopause); or (2) has undergone surgical sterilization (i.e., hysterectomy and/or bilateral oophorectomy for removal of uterus and/or ovaries). Individuals with any condition or social circumstance that, in the opinion of the investigator, would impair the participant's ability to comply with study activities, interfere with participant safety, or study endpoints.
7. Diagnosis of myelodysplastic syndrome (MDS) or acute myeloid leukemia (AML).
8. Taking a prohibited medication that cannot be safely discontinued or substituted.
