Inclusion Criteria:

          -  Phase 1 and Phase 2: confirmed diagnosis of previously treated relapsed and/or
             refractory lymphoma; patients with central nervous system (CNS) lymphoma are included

          -  Patient with leukemia phase (peripheral blood involvement), CNS lymphoma [including
             cerebrospinal fluid (CSF)-only disease], non-measurable disease, gastrointestinal (GI)
             mantle cell lymphoma (MCL), or bone marrow (BM) MCL are also eligible;
             gastrointestinal or bone marrow or spleen only patients are allowable and will be
             analyzed separately

          -  All adverse events related to prior therapies (chemotherapy, radiotherapy, and/or
             surgery) must be resolved to =< grade 1, except for alopecia

          -  Patients must be willing to receive transfusions of blood products

          -  Eastern Cooperative Oncology Group (ECOG) performance status of 2 or less

          -  Serum creatinine < 2.0 mg/dl

          -  Serum bilirubin < 1.5 mg/dl

          -  Platelet count > 50,000/mm^3

          -  Absolute neutrophil count (ANC) > 1,000/mm^3

          -  Alanine aminotransferase (ALT), or aspartate aminotransferase (AST) < 2 x upper limit
             of normal or < 5 x upper limit of normal if hepatic metastases are present

          -  Willing and able to participate in all study related procedures and therapy including
             swallowing capsules without difficulty

          -  Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy
             test and must be willing to use acceptable methods of birth control during the study
             and for 90 days after the last dose of study treatment; acceptable methods of birth
             control include condoms with birth control foam, birth control pills, implantable or
             injectable birth control, birth control patch, intrauterine device (IUD), or diaphragm
             with spermicidal gel; male patients must use an effective barrier method of
             contraception (i.e. , condoms with birth control foam or diaphragm with spermicidal
             gel) during the study and for 90 days following the last dose of study treatment if
             sexually active with a female of childbearing potential; contraception must be in
             place at least 2 weeks prior to initiating study treatment; a female of childbearing
             potential is a sexually mature woman who: 1) has not undergone a hysterectomy or
             bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24
             consecutive months (i.e., has had menses at any time in the preceding 24 consecutive
             months)

          -  Patient must be English-speaking [MD Anderson Symptom Inventory (MDASI) completion
             only]

        Exclusion Criteria:

          -  Any serious medical condition including but not limited to, uncontrolled hypertension,
             uncontrolled diabetes mellitus, uncontrolled infection, active/symptomatic coronary
             artery disease, chronic obstructive pulmonary disease (COPD), renal failure, active
             hemorrhage, or psychiatric illness that, in the investigators opinion places the
             patient at unacceptable risk or would prevent the subject from signing the informed
             consent form

          -  Pregnant or breast feeding females

          -  Use of any standard/experimental anti-lymphoma drug therapy, including steroids
             (dexamethasone dose >= 4 mg/day or prednisone >= 20 mg/day), within 3 weeks of
             initiation of the study or use of any experimental non-drug therapy (e.g., donor
             leukocyte/mononuclear cell infusions) within 56 days of initiation of the study drug
             treatment; hydroxyurea is permitted up to 24 hours before the first dose of study drug
             in patients with rapidly-proliferating disease

          -  Prior allogeneic stem cell transplant (SCT) within 16 weeks or autologous SCT within 8
             weeks of initiation of therapy (patients that require immunosuppressive therapy are
             not eligible within 60 days of therapy)

          -  History of human immunodeficiency virus (HIV) infection; patients with active
             hepatitis B infection (not including patients with prior hepatitis B vaccination; or
             positive serum hepatitis B antibody); hepatitis C infection is allowed as long as
             there is no active disease and is cleared by GI consultation; HIV screening is not
             required for this study

          -  Significant neuropathy (grades 3-4, or grade 2 with pain) within 14 days prior to
             enrollment

          -  Malabsorption syndrome, disease significantly affecting gastrointestinal function, or
             resection of the stomach or small bowel or ulcerative colitis, symptomatic
             inflammatory bowel disease, or partial or complete bowel obstruction, or any other
             gastrointestinal condition that could interfere with the absorption and metabolism of
             ONC201

          -  Major surgery within 4 weeks of initiation of therapy

          -  The patient has a prior or concurrent malignancy that in the opinion of the
             investigator, presents a greater risk to the patient's health and survival, than of
             the MCL, within the subsequent 6 months at the time of consent; investigator
             discretion is allowed

          -  Patients with New York Heart Association (NYHA) class III and IV heart failure,
             myocardial infarction in the preceding 6 months, and significant conduction
             abnormalities, including but not limited to second (2nd) degree atrioventricular (AV)
             block type II, third (3rd) degree block, QT prolongation (corrected QT [QTc] > 500
             msec), sick sinus syndrome, ventricular tachycardia, symptomatic bradycardia (heart
             rate < 50 beats per minute [bpm]), hypotension, light headedness and syncope; patients
             with active atrial fibrillation will be excluded; the protocol excludes patients who
             have within the past year had a stent and by recommendation of their cardiologist need
             to stay on anticoagulants such as warfarin equivalent vitamin K antagonist

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to ONC201 or its excipients

          -  Acute infection requiring treatment (systemic antibiotics, antivirals, or antifungals)
             within 14 days prior to initiation of study

          -  Active alcoholism or use of recreational drug (evaluated by history taking)