A Study of LY2606368 (Prexasertib) in Patients With Solid Tumors With Replicative Stress or Homologous Repair Deficiency

Inclusion Criteria: * Participants must have a pathologically confirmed advanced solid tumor for which standard therapy proven to provide clinical benefit does not exist or is no longer effective. * Participants must have one of the following (confirmed via targeted NextGeneration sequencing \[NGS\] using the DFCI/BWH OncoPanel or another CLIA-certified method): * For the replicative stress cohort: MYC amplification, CCNE1 amplification, Rb loss, FBXW7 mutation, or another genomic abnormality indicative of replicative stress as agreed upon with the principal investigator. OR * For the HR deficiency cohort: genomic or somatic mutation in BRCA1, BRCA2, PALB2, RAD51C, RAD51D, ATR, ATM, CHK2, the Fanconi anemia pathway genes, or another genomic or somatic mutation in a known HR gene as agreed upon with the principal investigator. * For the CCNE1 cohort: CCNE1 amplification of 6-fold or greater. Patients with borderline amplification levels may be considered following approval from the overall principal investigator. * Participants must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm with conventional techniques or as ≥ 10 mm with spiral CT scan, MRI, or calipers by clinical exam. * Age ≥ 18 years. * ECOG performance status \< 2 * Participants must have adequate organ and marrow function as defined below: * Absolute neutrophil count ≥ 1.5 K/uL * Platelet count ≥ 100 K/uL * Hemoglobin ≥ 9 g/dL (with or without transfusion support) * Total bilirubin ≤ 1.5 × institutional upper limit of normal (ULN) * AST(SGOT)/ALT(SGPT) ≤ 2.5 × institutional ULN, unless liver metastases are present and then ≤ 5 × institutional ULN is acceptable * Serum creatinine ≤ 1.5 × institutional ULN * Participants enrolling to the HR or replicative stress cohorts during Stage 1 must have disease that is amenable to biopsy and be willing to undergo a pre-treatment tumor biopsy. * The potential effects of LY2606368 use during pregnancy and lactation are not known. Nonclinical studies of LY2606368 on pregnancy and fetal development have not been performed. To minimize any potential risks, men and women with reproductive potential should use medically approved contraceptive precautions during treatment and for 3 months following the last dose of LY2606368. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and for 3 months after completion of LY2606368 administration. * Ability to understand and the willingness to sign a written informed consent document. * QTcF value of ≤ 470 msec on screening electrocardiogram (EKG) Exclusion Criteria: * Participants who have had chemotherapy, other investigational or biologic therapy, major surgery, or radiotherapy within 3 weeks (6 weeks for nitrosoureas or mitomycin C) prior to the planned first dose of LY2606368 therapy. * Participants who have not recovered to eligibility levels from prior toxicity or adverse events as a result of previous treatment prior to the study. * Participants who have received prior treatment with a CHK1 inhibitor. * Participants who have received prior radiation therapy to \> 25% of the bone marrow. * Participants with known untreated brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. Participants with a history of brain metastases that have been treated, are no longer taking corticosteroids, and have been stable on imaging for at least one month following the end of treatment are permitted. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to LY2606368. * Participants with a personal or family history of long QT syndrome. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, NYHA Class III/IV heart failure, unstable angina pectoris, cardiac arrhythmia, myocardial infarction within 3 months of enrollment, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant or breastfeeding females. The potential effects of LY2606368 use during pregnancy and breastfeeding are not known and LY2606368 has the potential for teratogenic or abortifacient effects. * Known HIV-positive participants are ineligible because these participants are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in HIV-positive participants when indicated. * Participants enrolling to the HR or replicative stress cohort during Stage 1 may not be on anticoagulant therapy unless the treating investigator has deemed it safe to temporarily hold to facilitate the pre-treatment tumor biopsy.