The goal of this clinical research study is to find the highest tolerable dose of vorinostat that can be given with gemcitabine, busulfan, and melphalan with a stem cell transplant. Researchers also want to learn about the safety and level of effectiveness of this combination.

Busulfan and melphalan are designed to kill cancer cells by binding to DNA (the genetic material of cells), which may cause cancer cells to die.

Gemcitabine is designed to disrupt the growth of cancer cells, which may cause cancer cells to die. It may help to increase the effect of busulfan and melphalan on cancer cells by not allowing these cells to repair the DNA damage caused by busulfan or melphalan.

Vorinostat is designed to open up the DNA and allow greater access to drugs that bind to DNA, such as gemcitabine, busulfan and melphalan.

Study Groups:

If you are found to be eligible to take part in this study, you will be enrolled in a group of at least 3 participants to begin receiving the study drugs.

The dose of the study drugs you receive will depend on when you enrolled in this study. If no intolerable side effects occur in your group, researchers will continue to enroll participants at the next dose level until either the vorinostat reaches the dose level currently used alone without stem cell transplant, or the highest tolerable dose of this drug is found. The dose that you receive will remain the same throughout this study.

Busulfan Test Dose:

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 levels changes over time. This information will be used to decide the next dose needed to reach the target blood 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 it cannot be given 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.

About 11 samples of blood (about 1 teaspoon each time) will be drawn for pharmacokinetic (PK) testing 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. These blood samples will be drawn at various timepoints before you receive busulfan and over about the next 11 hours. 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 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.

If you receive the busulfan test dose as an outpatient:

On Days -12, -11, and -10, 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.

You will be admitted to the hospital on Day -9.

If you receive the busulfan test dose as an inpatient:

On Days -13, -12, and -11, 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.

You will be admitted to the hospital on Day -11.

Study Drug Administration (for all patients):

In stem cell transplant, 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.

Beginning on Day -9, you will swish the liquids 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. These drugs are used to help decrease the risk of side effects in the mouth and throat.

On Day -8 through Day -2, you will take vorinostat by mouth, with food.

On Day -8, you will receive gemcitabine by vein over 3 1/2 - 4 1/2 hours and busulfan by vein over 3 hours.

On Days -8, -7, -6, and -5, you will receive busulfan by vein over 3 hours.

On Day -3, you will receive gemcitabine by vein over 3 1/2 - 4 1/2 hours and melphalan by vein over 30 minutes.

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

On Day -1, you will rest.

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.

If you have a B-cell cancer, you will receive rituximab (a treatment used for certain lymphomas or chronic lymphocytic leukemia) by vein over 3-6 hours as part of standard of care, on Days +1 and +8.

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

Study Tests:

On Day -1, you will have an electrocardiogram (ECG) to check your heart function.

About 30-100 days after the transplant, you will have lung function tests.

About 100 days after the transplant:

• Blood (about 4 teaspoons) will be drawn for routine tests.
• If the doctor thinks it is needed, you may have a bone marrow aspiration and biopsy to check the status of the disease.
• You will have a CT scan of the neck, chest, abdomen, and pelvis to check the status of the disease.
• If the doctor thinks it is needed, you will have a PET scan to check the status of the disease.

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, or if you are unable to follow study directions.

This is an investigational study. Vorinostat, gemcitabine, busulfan, melphalan, and rituximab are all FDA approved and commercially available. The use of these study drugs in combination is investigational.

Up to 80 patients will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Age 12 to 65 years
2. Patients with primary refractory or recurrent non-Hodgkin's lymphoma (NHL) or HL that do not qualify for treatment protocols of higher priority.
3. Patients with double-hit NHL, in any state of the disease.
4. Patients with peripheral T-cell lymphoma not otherwise specified (PTCL-NOS) in any state of the disease.
5. Angioimmunoblastic T-cell lymphoma (AITL) in any stage of the disease.
6. Adequate renal function, as defined by estimated serum creatinine clearance >/=50 ml/min and/or serum creatinine </= 1.8 mg/dL.
7. Adequate hepatic function, as defined by serum glutamate oxaloacetate transaminase (SGOT) and/or serum glutamate pyruvate transaminase (SGPT) </= 3 x upper limit of normal; serum bilirubin and alkaline phosphatase </= 2 x upper limit of normal.
8. Adequate pulmonary function with forced expiratory volume at one second (FEV1), forced vital capacity (FVC) and diffusing capacity of lung for carbon monoxide (DLCO) >/= 50% of expected corrected for hemoglobin.
9. Adequate cardiac function with left ventricular ejection fraction >/= 40%. No uncontrolled arrhythmias or symptomatic cardiac disease.
10. Zubrod performance status <2.
11. Negative Beta diffusing capacity of lung for carbon monoxide (HCG) text in a woman with child-bearing potential, defined as not post-menopausal for 12 months or no previous surgical sterilization

Exclusion Criteria:

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