Inclusion Criteria:

1. Patients must have a new diagnosis of T-cell NHL eligible histologies include:Peripheral T-cell lymphoma (unspecified), CD 30 + anaplastic large cell lymphoma ( ALK-1 positive and ALK-1 negative), angioimmunoblastic T-cell lymphoma, intestinal T-cell lymphoma, subcutaneous panniculitic T-cell lymphoma.
2. Patients who are eligible for blood and marrow transplant can receive this treatment to maximal reduction of tumor bulk. A minimum of four cycles of therapy will be given before evaluation for to hematopoetic stem cell transplant.
3. Patients must have at least one clear-cut bi-dimensionally measurable site (> 10mm by spiral computerized tomography - CT - scan or > 20mm by conventional measures). Baseline measurements of measurable sites and evaluation of evaluable disease must be obtained within four weeks prior to registration of this study.
4. Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2.
5. Patients must be age 18 years old and above.There is no dosing or adverse event data are currently available on the use of vorinostat in patients <18 years of age, children are excluded from this study but may be eligible for future pediatric phase 2 combination trials.
6. Patients are required to have adequate bone marrow reserve as indicated: Absolute neutrophil count (ANC) >/= 1000/mm^3, Platelets >/= 80,000/mm^3, Hemoglobin >/= 8g/dL. If there is bone marrow involvement by lymphoma then there is no minimum level of counts required. These values must be obtained within two weeks before protocol entry.
7. Patients must have adequate liver function as indicated by: Bilirubin </= 1.5 times the upper limit of normal (ULN), Alanine transaminase (ALT) </=2 times the (ULN) or aspartate transaminase (AST) </= 2 times the ULN. These values must be obtained within two weeks before protocol entry.
8. Patients are required to have adequate renal function as indicated by a serum creatinine </= 2.5 mg/dL. This value must be obtained within two weeks before protocol entry.
9. Left ventricular ejection fraction must be evaluated by nuclear medicine scan or echocardiography and measure >/= 50%.
10. Concomitant steroids may continue provided they have been on a stable dosage for at least 3 months prior to enrollment. Patients may continue to receive systemic corticosteroids as premedications for chemotherapy, antibiotics, and transfusions.
11. Male patients must agree to use a barrier method of contraception or agree to abstain from heterosexual activity for the duration of the study
12. Female patients must be willing to use two adequate barrier methods of contraception to prevent pregnancy or agree to abstain from heterosexual activity throughout the study or be post menopausal (free from menses > two years or surgically sterilized).
13. Female patients of childbearing potential must have a negative serum pregnancy test (Beta hCG) within 72 hours of receiving the first dose of vorinostat.
14. Patients must have the ability able to give informed consent.

Exclusion Criteria:

1. 1. Patients with a) T-cell lymphoma with skin involvement only are excluded if they have no evidence of systemic disease b)T-cell prolymphocytic leukemia (T-PLL) c) T-cell large granular lymphocytic leukemia d) Primary cutaneous CD30+ disorders: anaplastic large cell lymphoma and lymphomatoid papulosis e) Angiocentric/nasal type T/NK-cell lymphoma f) Hepatosplenic gamma-delta T-cell lymphoma
2. Patients with active Hepatitis B and/or Hepatitis C infection.
3. Patients with known HIV infection are excluded. a) These patients are excluded secondary to potential to target activated T-cells, in a population of patients already at risk for T-cell depletion, would be a contraindication to therapy.
4. Patients with active infections requiring specific anti-infective therapy are not eligible until all signs of infections are resolved.
5. Patients with pre-existing cardiovascular disease requiring ongoing treatment. This includes:a) Congestive heart failure, b) Severe CAD, c) Cardiomyopathy, d) Uncontrolled cardiac arrhythmia, e) Unstable angina pectoris, f) Recent MI (within 6 months).
6. Patients with prior exposure to either vorinostat (including other HDAC inhibitors except valproic acid) or anthracyclines: a) Patients who have received valproic acid (VPA) for the treatment of seizures may be enrolled on this study, but must not have received VPA within 30 days of study enrollment.
7. Patients who are pregnant or breast-feeding. a)Effects of this treatment on the fetus and young children are unknown at this time.
8. Patients who have had an invasive solid tumor malignancy in the past five years except non-melanoma skin cancers or cervical carcinoma in situ or ductal/lobular carcinoma in situ of the breast who is currently without evidence of disease.
9. Patients undergoing anti-neoplastic chemotherapy, radiation, hormonal (excluding contraceptives) or immunotherapy, or investigational medications within the past four weeks.
10. Patients with deep vein thrombosis within three months.
11. Patient with concurrent use of complementary or alternative medicines.
12. Patients with psychiatric illness and/or social situations that would limit compliance with the study medication and requirements.
13. Patients with grade 2 or more neuropathy.
14. Patients with known CNS lymphoma.