Inclusion Criteria:

          -  Baseline evaluations are to be conducted within 14 days prior to start of protocol
             therapy, with the exception of the informed consent and baseline tumor imaging which
             may be obtained up to 28 days prior to the start of protocol therapy.

          -  For enrollment into the MET cohort: Participants must have a histologically or
             cytologically confirmed advanced NSCLC and must have received at least one prior line
             of therapy in the metastatic setting.

          -  For enrollment into the NTRK cohort: Participants must have a histologically or
             cytologically confirmed advanced solid tumor and must have received at least one prior
             line of therapy in the metastatic setting.

          -  Participants enrolling into the MET cohort must have a MET exon 14 mutation as
             confirmed by targeted NextGen Sequencing using the DFCI/BWH OncoPanel or another
             CLIA-certified method. Participants whose NSCLC specimens contain actionable genetic
             mutations/alterations (e.g. ALK/EGFR) should receive appropriate targeted therapies
             prior to enrollment in the trial.

          -  Participants enrolling into the NTRK cohort must have an NTRK1, 2, or 3 rearrangement
             as confirmed by targeted NextGen Sequencing using the DFCI/BWH OncoPanel or another
             CLIA-certified method.

          -  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.

          -  Participants enrolling to the MET cohort who have received treatment with a prior MET
             inhibitor must be able and willing to undergo a baseline tumor biopsy.

          -  Participants enrolling to the NTRK cohort who have received treatment with a prior
             NTRK inhibitor must be able and willing to undergo a baseline tumor biopsy.

          -  Age ≥ 18 years. As no dosing or adverse event data are currently available in
             participants < 18 years of age, children are excluded from this study but will be
             eligible for future pediatric trials.

          -  ECOG performance status ≤ 1 (see Appendix A).

          -  Participants must have normal organ and marrow function as defined below:

               -  Absolute neutrophil count ≥ 1.5 K/uL

               -  Platelets ≥ 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

          -  The effects of merestinib on the developing human fetus are unknown. For this reason
             and because MET inhibitor agents are known to be teratogenic, women of child-bearing
             potential and men must agree to use adequate contraception (hormonal or barrier method
             of birth control; abstinence) prior to study entry and for the duration of study
             participation. 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 4
             months after completion of merestinib administration.

          -  Ability to understand and the willingness to sign a written informed consent document.

          -  Participants must be able to swallow and retain oral medication or have an enteral
             feeding tube in place for study drug administration.

          -  Participants who have received prior oral tyrosine kinase inhibitors (TKIs) will be
             allowed on study if at least 5 half-lives have elapsed since the date of their last
             dose of TKI.

        Exclusion Criteria:

          -  Participants who have had chemotherapy, immune therapy, other investigational therapy,
             major surgery, or radiotherapy within 3 weeks (6 weeks for nitrosoureas or mitomycin
             C) prior to entering the study.

          -  Participants who have not recovered to eligibility levels from adverse events due to
             agents administered more than 3 weeks earlier.

          -  Participants who are receiving any other investigational agents.

          -  Participants with 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 ≥ 4
             weeks following the last date of treatment are permitted. Note: a brain MRI is
             required during the screening period.

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to merestinib.

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements.

          -  Pregnant women are excluded from this study because merestinib is an agent 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 merestinib, breastfeeding should be discontinued if the mother is treated
             with merestinib.

          -  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 participants receiving combination
             antiretroviral therapy when indicated.

          -  Participants enrolling to the MET cohort who have received a prior MET inhibitor may
             not be on anticoagulant therapy unless the investigator has deemed it safe to
             temporarily hold to facilitate the baseline tumor biopsy.

          -  Participants enrolling to the NTRK cohort who have received a prior NTRK inhibitor may
             not be on anticoagulant therapy unless the investigator has deemed it safe to
             temporarily hold to facilitate the baseline tumor biopsy.

          -  Participants with uncontrolled high blood pressure, defined as a blood pressure during
             screening of ≥ 160/100 despite medical management.

          -  Participants must not have any clinically significant gastrointestinal abnormalities
             that in the opinion of the treating investigator may alter absorption of oral
             medications, such as malabsorption syndrome or major resection of the stomach or
             bowels.

          -  Participants with a history of a second primary malignancy. Exceptions include:
             patients with a history of malignancies that were treated curatively and have not
             recurred within 3 years prior to study entry; resected basal and squamous cell
             carcinomas of the skin, and completely resected carcinoma in situ of any type.