1. Age: Patients must be >= 12 months and <= 21 years of age at the time of study enrollment. 2. Diagnosis: * Leukemia Patients must have relapsed or refractory acute lymphoblastic leukemia (ALL) with: a. ALL >= 25% blasts in the bone marrow (M3) and any central nervous system (CNS) status. OR b. ALL with an M2 marrow (>= 5% to < 25% blasts) with an extramedullary site of relapse; including CNS 2 or CNS 3. c. Refractory disease defined as no more than 1 prior failed salvage attempt following the current relapse, or no more than 2 additional treatment cycles after initial induction failure in newly diagnosed patients. * Lymphoma Patient must have relapsed or refractory lymphoma with: a. Lymphoblastic lymphoma or peripheral T-cell lymphoma. b. Histologic verification of disease at original diagnosis or subsequent relapse. c. Evaluable or measurable disease documented by clinical or radiographic criteria or bone marrow disease present at study entry. d. Patient may have CNS 2 or 3 status if other sites of leukemia or lymphoma involvement are present. 3. Performance Level: Karnofsky >= 50% for patients > 16 years of age and Lansky >= 50 for patients <= 16 years of age. 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. 4. Prior Therapy: Patients must have fully recovered from the acute toxic effects of all prior anti-cancer chemotherapy, defined as resolution of all such toxicities to <= Grade 2 or lower per the inclusion/exclusion criteria. a. Patients must have had either: * Relapse after going into remission from re-induction for the first or subsequent relapse (ie: 2nd, 3rd, 4th[?]relapse), * Patients with lymphoma may have refractory disease after first or greater relapse and a single re-induction attempt. b. Myelosuppressive chemotherapy: * Patients with leukemia or lymphoma who relapse while receiving maintenance chemotherapy will not be required to have a waiting period before enrollment onto this study. * At least 14 days must have elapsed after the completion of cytotoxic therapy, with the exception of hydroxyurea. Note: Cytoreduction with hydroxyurea in patients can be initiated and continued for up to 24 hours prior to the start of protocol therapy. * Patients who received intrathecal chemotherapy within 7 days of therapy initiation remain eligible, but should not receive the day 1 intrathecal (IT) chemotherapy. c. Hematopoietic growth factors: At least 14 days after the last dose of a long-acting growth factor (e.g. Neulasta) or 7 days for short-acting growth factor (e.g., Filgrastim). d. Biologic (anti-neoplastic agent): At least 7 days after the last dose of a biologic agent. For agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur. The duration of this interval must be discussed with the study chair. e. Immunotherapy: At least 42 days after the completion of any type of immunotherapy, e.g. tumor vaccines or chimeric antigen receptor T cell (CART) therapy f. Monoclonal antibodies: At least 3 half-lives of the antibody must have elapsed after the last dose of a monoclonal antibody with the exclusion of blinatumomab. (ie: Rituximab = 66 days, Epratuzumab = 69 days). Patients must have been off blinatumomab infusion for at least 7 days and all drug-related toxicity must have resolved to grade 2 or lower as outlined in the inclusion and exclusion criteria. g. External Beam Radiation Therapy (XRT): At least 14 days after local palliative XRT (small port); At least 84 days must have elapsed if prior TBI, craniospinal XRT or if >= 50% radiation of pelvis; At least 42 days must have elapsed if other substantial marrow radiation. h. Stem Cell Infusion: No evidence of active graft vs. host disease and at least 84 days must have elapsed after transplant or stem cell infusion. i. Study-specific limitations on prior therapy: Patient may not have previously received therapy with an mTOR inhibitor. 5. Organ Function Requirements * Adequate Bone Marrow Function Defined as: - Patients should not be known to be refractory to red blood cell or platelet transfusions. - Blood counts are not required to be normal prior to enrollment on trial. However, platelet count must be >= 20,000/mm3 to initiate therapy (may receive platelet transfusions). * Adequate Renal Function Defined as: - Creatinine clearance or radioisotope glomerular filtration rate (GFR) >= 70ml/min/1.73 m2 or - A serum creatinine based on age/gender as indicated in the protocol. * Adequate Liver Function Defined as: - Total bilirubin (sum of conjugated + unconjugated) <= 1.5 x institutional upper limit of normal for age - Serum glutamic pyruvic transaminase (SGPT) (ALT) and serum glutamic-oxaloacetic transaminase (SGOT) (AST) must be less than 3 x institutional upper limit of normal (Grade 1 or less per Common Terminology Criteria for Adverse Events (CTCAE) 4). - Gamma-glutamyl transferase (GGT) must be less than 2.5 x institutional upper limit of normal (Grade 1 or less per CTCAE 4). - Serum albumin >= 2 g/dL. - The hepatic requirements may be waived for patients with Grade 1 or 2 elevations of hepatic transaminases clearly due to leukemic infiltration after consultation with a study Co-Chair. * Adequate Cardiac Function Defined As: - Shortening fraction of >= 27% by echocardiogram, or - Ejection fraction of >= 50% by gated radionuclide study/echocardiogram. * Pulmonary Function Defined as: - Pulse oximetry > 94% on room air (> 90% if at high altitude) - No evidence of dyspnea at rest and no exercise intolerance. - Baseline chest x-ray with no evidence of active infectious disease or pneumonitis. * Reproductive Function: - Female patients of childbearing potential must have a negative urine or serum pregnancy test confirmed prior to enrollment. - Female patients with infants must agree not to breastfeed their infants while on this study. - Male and female patients of child-bearing potential must agree to use an effective method of contraception approved by the investigator during the study. * Random or fasting glucose within the upper limits of normal for age. If the initial blood glucose is non-fasting and above normal limits a fasting glucose can be obtained and must be within the upper limits of normal for age. * Triglyceride/Cholesterol: Fasting or non-fasting serum triglyceride level <= 300 mg/dL and serum cholesterol level <= 300 mg/dL. 6. Informed Consent: All patients and/or their parents or legal guardians must sign a written informed consent. Age appropriate assent will be obtained per institutional guidelines. To allow non-English speaking patients to participate in this study, bilingual health services will be provided in the appropriate language when feasible. 7. Protocol Approval: All institutional, FDA, and NCI requirements for human studies must be met.