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The goal of this clinical research study is to learn about the safety and effectiveness of the combination of panobinostat, gemcitabine, busulfan, and melphalan, either with or without rituximab, in patients who have non-Hodgkin's or Hodgkin's lymphoma and are receiving a stem cell transplant.

Busulfan Test Dose:

Participant will receive a test dose of busulfan by vein over about 60 minutes. Before and after participant receives this low-level test dose of busulfan, blood will be drawn several times to check how the level of busulfan in participant's blood changes over time. This is called pharmacokinetic (PK) testing. This information will be used to decide the next dose needed to reach the correct blood level that matches participant's body size. Participant will most likely receive this test dose as an outpatient during the week before participant is admitted to the hospital. If it cannot be given as an outpatient, participant will be admitted to the hospital on Day -11 (11 days before participant's stem cells are returned to participant's body) and the test dose will be given on Day -10.

Blood (about 1 teaspoon each time) will be drawn for PK testing about 11 times before participant's test dose of busulfan and 10 times over the 11 hours after the dose. The blood samples will be repeated again on the first day of high-dose busulfan treatment (Day -8). A temporary heparin lock line will be placed in participant's 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 any reason, participant will receive the standard fixed dose of busulfan.

Study Drug Administration:

On Days -13 through -11 (the 3 days before participant is admitted to the hospital), participant will receive palifermin by vein over about 30 seconds each day to help decrease the risk of side effects in the mouth and throat.

On Day -9 through Day -2, participant will take panobinostat by mouth, with food. The study staff will tell participant how many tablets to take. Participant will receive dexamethasone by vein over about 3-5 minutes 2 times each day.

On Day -9, if participant has a type of B-cell cancer, participant will receive rituximab (a treatment used for certain lymphomas) by vein over about 3-6 hours as part of participant's standard care. The study doctor will tell participant if participant will receive rituximab.

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

On Days -8, -7, -6, and -5, participant will receive busulfan by vein over about 60 minutes.

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

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

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

On Day 0, participant will receive participant's stem cells by vein over about 30-60 minutes.

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

As part of standard care, participant will receive G-CSF (filgrastim) as an injection just under participant's skin 1 time each day starting on Day +5 until participant's blood cell levels return to normal. The study doctor will discuss this with participant, including how filgrastim is given and its risks.

Length of Study:

As part of standard care, participant will remain in the hospital for about 3-4 weeks after the transplant. After participant is released from the hospital, participant will continue as an outpatient. Participant will need to stay in the Houston area to be checked for infections and side effects.

Participant will be taken off study about 100 days after the transplant. Participant may be taken off study earlier if the disease gets worse, if intolerable side effects occur, or if participant is unable to follow study directions.

Study Tests:

About 100 days after the transplant:

• Participant will have a physical exam.
• Blood (about 4 teaspoons) and urine will be collected for routine tests.
• If the doctor thinks it is needed, participant may 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 is numbed with anesthetic, and a small amount of bone marrow and bone is withdrawn through a large needle.

This is an investigational study. Panobinostat and melphalan are FDA approved and commercially available for the treatment of multiple myeloma (MM). Busulfan is FDA approved and commercially available for the treatment of leukemia. Gemcitabine is FDA approved and commercially available for the treatment of lymphoma, breast cancer, and lung cancer. Rituximab is FDA approved and commercially available for the treatment of non-Hodgkin's lymphoma.

The use of these study drugs in combination is investigational. The study doctor can explain how the study drugs are designed to work.

Up to 75 participants will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Age 15-65
2. Patients with: DLBCL with one of the following: 1. Primary refractory (no CR to 1st line); 2. High-risk relapse (CR1 <6 months, secondary IPI >1 or high LDH); or, 3. Refractory relapse: No response (SD or PD) to >/= 1 line of salvage.
3. Hodgkin's with one of the following: 1. Primary refractory (no CR to 1st line or PD within 3 months); 2. High-risk relapse (CR1 <1 year, extranodal relapse or B symptoms); or, 3. Refractory relapse: No response (SD or PD) to >/= 1 line of salvage.
4. T-NHL with one of the following: 1. Primary refractory (</= CR to 1st line or relapse within 6 months); or, 2. Nonresponsive (SD/PD) to >/= 1 line of salvage.
5. Adequate renal function, as defined by estimated serum creatinine clearance >/= 50 ml/min and/or serum creatinine </= 1.8 mg/dL
6. Adequate hepatic function (SGOT and/or serum glutamate pyruvate transaminase (SGPT) </= 3 x upper limit of normal (ULN); bilirubin and ALP </= 2 x ULN
7. 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%
8. Adequate cardiac function with left ventricular ejection fraction >/= 40%. No uncontrolled arrhythmias or symptomatic cardiac disease
9. PS <2
10. Negative Beta human chorionic gonadotropin (HCG) in woman with child-bearing potential

Exclusion Criteria:

1. Grade >/= 3 non-hematologic toxicity from prior 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 enroll