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.