• Patients must be > 12 months and < or equal to 21 years of age at the time of study enrollment. 
• Diagnosis:  Patients must have histologic verification of malignancy at original diagnosis or at relapse, to include any of the following malignancies (no other histology is eligible); a) Neuroblastoma - measurable; b) Neuroblastoma - MIBG evaluable; c) Rhabdomyosarcoma; d) Osteosarcoma; e) Ewing Sarcoma/Peripheral PNET; f) Non-RMS soft tissue sarcoma; g) Hepatoblastoma; h) Malignant germ cell tumor; i) Wilms tumor; j) Acute lymphoblastic leukemia; k) Acute myelogenous leukemia; l) Rhabdoid malignancy. 
• Patients must have radiographically measurable disease (with the exception of neuroblastoma).  Measurable disease is defined as the presence of at least one lesion on MRI or CT scan that can be accurately measured with the longest diameter a minimum of 20 mm in at least one dimension.  Patients with neuroblastoma who do not have measurable disease but have MIBG+evaluable disease are eligible. 
• Disease status for Leukemia Patients:  Patients with leukemia must be recurrent or refractory to at least two prior induction or treatment regimens, in addition to the following criteria:  Acute lymphoid leukemia: > 25% blasts in the bone marrow (M3 bone marrow), excluding patients with known CNS disease.  Acute myeloid leukemia according to FAB classification - >= 5% blasts in the bone marrow; excluding patients with known CNS disease. 
• Patients must have a Lansky or Karnofsky performance status score of greater than or equal to 50, corresponding to ECOG categories 0, 1, or 2.  Use Karnofsky for patients > 16 years of age and Lansky for patients less than or equal to 16 years of age.  Note:  Patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. 
• Patients must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy or radiotherapy prior to study enrollment. 
• Adequate bone marrow function, adequate renal function, and adequate liver function. 

Exclusion Criteria: 

• Patients who are pregnant or breast-feeding are not eligible for this study as there is yet no available information regarding human fetal or teratogenic toxicities.  Negative pregnancy tests must be obtained in girls who are post-menarchal.  Males or females or reproductive potential may not participate unless they have agreed to use an effective contraceptive method for the duration of the study therapy.  MLN8237 may also potentially be secreted in milk and therefore breastfeeding women are excluded. 
• Growth factor(s):  Growth factors that support platelet or white cell number or function must not have been administered within the 7 days prior to enrollment (14 days if pegfilgrastim).  Corticosteroids: Patients requiring corticosteroids who have not been on a stable or decreasing dose of corticosteroid for the 7 days prior to enrollment are not eligible.  Investigational Drugs: Patients who are currently receiving another investigational drug are not eligible.  Anti-cancer Agents: Patients who are currently receiving other anticancer agents are not eligible.  Benzodiazepines: Use of daily benzodiazepine therapy excludes a patients from being eligible because of the protential bezodiazepine-like effects of MLN8237.  P-lycoprotein substrates; Patients who are currently receiving dogoxin, cyclosporine, tacrolimus, or sirolimus are not eligible.  See Appendix IX for a complete list of p-glycoprotein substrates that should be use with caution. 
• Patients who are unable to swallow tablets are not eligible. Patients who have an uncontrolled infection are not eligible.  Leukemia patients with CNS disease are not eligible.  Patients who in the opinion of the investigator may not be able to comply with the safety monitoring requirements of the study are not eligible.