INCLUSION CRITERIA for Patients on All Strata EXCEPT Stratum P

   - Patients must be < 22 years of age at time of diagnosis (e.g., eligible until 22nd
   birthday).

   - Histologic diagnosis of AT/RT or MRT as documented by the institutional pathologist
   with loss of INI1 or BRG1 expression in tumor cells confirmed by immunohistochemistry,
   or by molecular confirmation of tumor-specific biallelic SMARCB1/SMARCA4 loss/mutation
   if INI1/BRG1 immunohistochemistry is not available. For Stratum A participants,
   histologic confirmation of the diagnosis of AT/RT or MRT may be from the original
   diagnosis or at the time of recurrence/progression.

   - Patients must have adequate organ function (bone marrow, renal, liver), as defined in
   the protocol.

   - Female patients who are at least 10-years-old or are post-menarchal must have a
   negative serum or urine pregnancy test prior to enrollment.

   - Patients of childbearing or child fathering potential must be willing to use a
   medically acceptable form of birth control, which includes abstinence during study
   treatment and 12 months after the last dose of alisertib.

Inclusion Criteria for Stratum A Participants:

   - Patients with recurrent or progressive AT/RT/MRT (either CNS and/or extra-CNS) with
   radiographically measurable disease as defined by at least 1 lesion that can be
   measured in 2 dimensions or with tumor cells present in the CSF taken within 2 weeks
   prior to enrollment.

   - Performance status defined by Karnofsky or Lansky > 60 (except for patients with
   posterior fossa syndrome). Use Karnofsky for patients > 16 years and Lansky for
   patients < 16 years. Note: Patients who are unable to walk because of paralysis, but
   who are up in a wheelchair, will be considered to be ambulatory for the purpose of
   assessing the performance score.

   - Patient has fully recovered from the acute toxic effects of chemotherapy,
   immunotherapy, or radiation therapy prior to entering this study:

      - Myelosuppressive chemotherapy: Patient has not received myelosuppressive
      chemotherapy within 3 weeks of enrollment onto this study (4 and 6 weeks if prior
      temozolomide and nitrosourea, respectively).

      - Hematopoietic growth factors: At least 7 days must have elapsed since the
      completion of therapy with a growth factor. At least 14 days must have elapsed
      after receiving pegfilgrastim.

      - Biologic (anti-neoplastic agent): At least 7 days must have elapsed since
      completion of therapy with a biologic agent. For agents that have known adverse
      events occurring beyond 7 days after administration, this period prior to
      enrollment must be extended beyond the time during which adverse events are known
      to occur.

      - Monoclonal antibodies: At least 3 half-lives must have elapsed since prior
      therapy that included a monoclonal antibody (see Appendix I).

      - Radiation therapy: at least 3 months must have elapsed since any irradiation
      unless measurable disease progression occurs at a site separate from the
      irradiated area and the patient has recovered from toxicities associated with
      radiation therapy.

   - Patients with progressive synchronous/metachronous AT/RT and MRT will be eligible for
   stratum A3.

   - Patients may not have previously received alisertib.

   - Live expectancy >8 weeks.

   - Stable neurologic deficits on a stable dose of corticosteroids (if applicable) for at
   least 1 week before study enrollment.

Inclusion Criteria for Strata B or C Participants:

   - Patients with newly diagnosed AT/RT.

   - Performance status defined by Karnofsky or Lansky > 30 (except for patients with
   posterior fossa syndrome). Other requirements of performance evaluation are the same
   as for Stratum A participants.

   - No previous anticancer therapy (radiation therapy or chemotherapy) other than the use
   of corticosteroids.

   - Patients must begin treatment as outlined in the protocol within 42 days of definitive
   surgery (day of surgery is day 0; definitive surgery includes last surgery to resect
   residual tumor).

Inclusion Criteria for Stratum D Participants:

   - Patients with newly diagnosed AT/RT and synchronous extra-CNS MRT.

   - Performance status defined by Karnofsky or Lansky > 30 (except for patients with
   posterior fossa syndrome). Other requirements of performance evaluation are the same
   as for Stratum A participants.

   - No previous anticancer therapy (radiation therapy or chemotherapy) other than the use
   of corticosteroids.

   - Patients must begin treatment within 42 days of definitive surgery (day of surgery is
   day 0; definitive surgery includes repeat surgeries to resect residual tumor).

Inclusion Criteria for Stratum P Participants:

   - Biological parent of patient enrolling on the protocol (SJATRT) will be assigned to
   Stratum P.

EXCLUSION CRITERIA for All Strata Except Stratum P:

   - Clinically significant medical disorders that could compromise the ability to tolerate
   protocol therapy or that would interfere with the study procedures or results history.

   - Presence of an active, uncontrolled infection.

   - Known history of uncontrolled sleep apnea syndrome or other conditions that could
   result in excessive daytime sleepiness, such as severe chronic obstructive pulmonary
   disease or a requirement for supplemental oxygen.

   - Requirement for constant administration of proton-pump inhibitor, H2 antagonist, or
   pancreatic enzymes. Intermittent uses of antacids or H2 antagonists are allowed while
   patients are on dexamethasone as described in the protocol.

   - Inability to comply with the safety monitoring requirements of the study, as judged by
   the investigator.

   - Female participants of childbearing potential cannot be pregnant or breast-feeding.

   - Patients who are receiving other investigational drugs 14 or fewer days before
   enrollment.

   - Treatment with clinically significant enzyme inducers, such as the enzyme-inducing
   antiepileptic drugs phenytoin, carbamazepine or phenobarbital, or with rifampin,
   rifabutin, rifapentine, or St. John's wort within 7 days prior to initiation of
   alisertib.

   - Known gastrointestinal disease or procedures that could interfere with the oral
   absorption or tolerance of alisertib. Examples include, but are not limited to partial
   gastrectomy, history of small intestine surgery, and celiac disease.

   - Myocardial infarction within 6 months prior to enrollment or New York Heart
   Association Class III or IV heart failure, uncontrolled angina, severe uncontrolled
   ventricular arrhythmias, or electrocardiographic evidence of acute ischemia or active
   conduction system abnormalities. If for some reason an electrocardiogram is obtained
   before study enrollment, any abnormalities detected should be documented as clinically
   irrelevant.

   - Other severe acute or chronic medical or psychiatric condition, including uncontrolled
   diabetes, malabsorption, resection of the pancreas or upper small-bowel, or
   requirement for pancreatic enzymes, any condition that would modify the absorption of
   oral medications in the small bowel or any laboratory abnormality that may increase
   the risk associated with study participation or investigational product administration
   or that may interfere with the interpretation of study results and, in the judgment of
   the investigator, would make the patient inappropriate for enrollment in this study.