The goal of this clinical research study is to find the highest tolerable dose of belinostat that can be given in combination with azacitidine, gemcitabine, busulfan, and melphalan to patients who are scheduled to have a stem cell transplant. If you have diffuse large B-cell lymphoma (DLBCL), you will also receive rituximab. Researchers also want to learn about the safety and effectiveness of this combination.

Study Groups:

If you are found to be eligible to take part in this study, you will be assigned to a dose level of belinostat based on when you join this study. Up to 5 dose levels of belinostat will be tested. At least 2 participants will be enrolled at each dose level. The first group of participants will receive the lowest dose level. Each new group will receive a higher dose than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of belinostat is found.

All participants will receive the same dose level of azacitidine, gemcitabine, busulfan, melphalan. All patients with DLBCL will also receive the same dose of rituximab. However, if the first group has intolerable side effects, the dose level of gemcitabine may be lowered for all other groups.

Busulfan Test Dose:

In stem cell transplants, the days before you receive your stem cells are called minus days. The day you receive the stem cells is called Day 0. The days after you receive your stem cells are called plus days.

You will receive a test dose of busulfan by vein over about 60 minutes. This low-level test dose of busulfan is to check how the level of busulfan in your blood level changes over time. This information will be used to decide the next dose needed to reach the dose level that matches your body size. You will most likely receive this as an outpatient during the week before you are admitted to the hospital. If busulfan cannot be given to you as an outpatient, you will be admitted to the hospital on Day -11 (11 days before your stem cells are returned to your body) and the test dose will be given on Day -10.

On either Day -11 or -10, blood (about 1 teaspoon each time) will be drawn for pharmacokinetic (PK) testing 11 times over the 11 hours after your test dose of busulfan. PK testing measures the amount of study drug in the body at different time points and will help the study doctor determine what your dose of busulfan should be on study. The PK blood draws will be repeated again on the first day of high-dose busulfan treatment (Day -8). A temporary heparin lock line will be placed in your vein to lower the number of needle sticks needed for these draws. If it is not possible for the PK tests to be performed for technical or scheduling reasons, you will receive the standard fixed dose of busulfan.

About 2 or 3 days before receiving the test dose of busulfan (depending on if you are an inpatient or outpatient), you will receive palifermin by vein over about 30 seconds each day to help decrease the risk of side effects in the mouth and throat.

Study Drug Administration (all patients):

Beginning on Day -8, you will swish caphosol and glutamine in your mouth 4 times a day for about 2 minutes each time. You will swish these liquids every day until you leave the hospital. You will swallow the glutamine. These drugs are used to help decrease the risk of side effects in the mouth and throat.

On Day -9 through Day -2, you will receive belinostat continuously by vein. You will also receive azacitidine by vein on each of these days over about 30 minutes. Depending on the type of cancer you have, you will also receive rituximab by vein over 3-6 hours as part of standard care on Day -9. The study staff will tell you if you will receive rituximab.

On Day -8, you will receive gemcitabine by vein over 4½ hours.

On Days -8, -7, -6, and -5, you will receive busulfan by vein over 2 hours.

On Day -3, you will receive gemcitabine by vein over 4½ hours and then melphalan by vein over 30 minutes.

On Day -2, you will receive melphalan by vein over 30 minutes.

On Day -1, you will rest (you will not receive chemotherapy).

On Day 0, you will receive your stem cells by vein over about 30-60 minutes.

You will receive 3 more doses of palifermin by vein over 15-30 seconds on Days 0, +1, and +2.

As part of standard care, you will receive G-CSF (filgrastim) as an injection just under your skin 1 time each day starting on Day +5 until your blood cell levels return to normal.

Length of Study:

As part of standard care, you will remain in the hospital for about 3-4 weeks after the transplant. After you are released from the hospital, you will continue as an outpatient in the Houston area to be monitored for infections and transplant-related complications.

You will be taken off study about 100 days after the transplant. You may be taken off study early if the disease gets worse, if intolerable side effects occur, if you are unable to follow study directions, or if you choose to leave the study early.

If for any reason you want to leave the study early, you must talk to the study doctor. It may be life-threatening to leave the study after you have started to receive the study drugs but before you receive the stem cell transplant because your blood cell counts will be dangerously low.

Follow-Up:

About 100 days after the transplant:

• You will have a physical exam.
• Blood (about 4 teaspoons) and urine will be collected for routine tests and to check your kidney and liver function.
• If the doctor thinks it is needed, you will have a computed tomography (CT) and/or positron emission tomography (PET) scan to check the status of the disease.
• If the doctor thinks it is needed, you will have a bone marrow aspiration and biopsy to check the status of the disease. To collect a bone marrow aspiration/biopsy, an area of the hip or other site is numbed with anesthetic, and a small amount of bone marrow and bone is withdrawn through a large needle.

This is an investigational study. Belinostat, busulfan, and rituximab are FDA approved and commercially available for the treatment of lymphoma. Gemcitabine is FDA approved and commercially available for the treatment of breast cancer, non-small cell lung cancer (NSCLC), ovarian cancer, and pancreatic cancer. Melphalan is FDA approved and commercially available for the treatment of multiple myeloma (MM). Azacitidine is FDA approved and commercially available for the treatment of myelodysplastic syndrome (MDS). The use of these study drugs in combination to treat lymphoma is considered investigational.

Up to 60 participants will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Age 15-65
2. Patients with: 2. 1. DLBCL with one of the following: 2.1.1. Primary refractory (no CR to 1st line). 2.1.2. High-risk relapse (CR1 <6 months, secondary IPI >1, high LDH). 2.1.3. Refractory relapse: No response (SD or PD) to >/= 1 line of salvage. 2.2. Hodgkin's with one of the following: 2.2.1. Primary refractory (no CR to 1st line or PD within 3 months). 2.2.2. High-risk relapse (CR1 <1 year, extranodal relapse, B symptoms). 2.2.3. Refractory relapse: No response (SD or PD) to >/= 1 line of salvage. 2.3. T-NHL with one of the following: 2.3.1. Primary refractory (</= CR to 1st line or relapse within 6 months). 2.3.2. Nonresponsive (SD/PD) to >/= 1 line of salvage. 2.4. Burkitt's or lymphoblastic lymphoma with one of the following: 2.4.1. Primary refractory (</= CR to 1st line or relapse within 6 months). 2.4.2. Refractory to at least 1 line of salvage (SD/PD).
3. Adequate renal function, as defined by estimated serum creatinine clearance >/= 50 ml/min and/or serum creatinine </= 1.8 mg/dL.
4. Adequate hepatic function (SGOT and/or serum glutamate pyruvate transaminase (SGPT) </= 3 x upper limit of normal (ULN); bilirubin and ALP </= 2 x ULN.
5. Adequate pulmonary function with forced expiratory volume at one second (FEV1), forced vital capacity (FVC) and diffusing capacity of lung for carbon monoxide (DLCO) (corrected for Hgb) >/= 50%.
6. Adequate cardiac function with left ventricular ejection fraction >/= 40%. No uncontrolled arrhythmias or symptomatic cardiac disease.
7. PS <2.
8. Negative Beta human chorionic gonadotropin (HCG) in woman with child-bearing potential.

Exclusion Criteria:

1. Grade >/= 3 non-hematologic toxicity from previous therapy that has not resolved to </= G1.
2. Prior whole brain irradiation.
3. Corrected QT interval (QTc) longer than 500 ms.
4. Active hepatitis B, either active carrier (HBsAg +) or viremic (HBV DNA >/= 10,000 copies/mL, or >/= 2,000 IU/mL).
5. Evidence of either cirrhosis or stage 3-4 liver fibrosis in patients with chronic hepatitis C or positive hepatitis C serology.
6. Active infection requiring parenteral antibiotics.
7. HIV infection, unless receiving effective antiretroviral therapy with undetectable viral load and normal cluster of differentiation 4 (CD4) counts.
8. Radiation therapy in the month prior to enrollment.