Brief Summary: This phase I trial studies the side effects and best dose of alisertib and romidepsin in treating patients with B-cell or T-cell lymphomas that have returned after a period of improvement (relapsed) or have not responded to treatment (refractory). Alisertib and romidepsin may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth.

Detailed Description: PRIMARY OBJECTIVES: I. To assess the safety profile of alisertib (MLN8237) plus romidepsin. II. To determine the maximum tolerated dose (MTD), if reached, of MLN8237 administered in combination with romidepsin. SECONDARY OBJECTIVES: I. To evaluate objective response rate (ORR) and complete response (CR) of the combined regimen. II. To assess whether higher levels of expression of aurora kinase A correlate with outcomes. III. To determine if this combination results in downregulation of targets of v-myc myelocytomatosis viral oncogene homolog (avian) (C-Myc) in C-Myc positive patients, induces mitotic catastrophe, changes immune system or other host responses, or upregulates markers for apoptosis. OUTLINE: This is a dose-escalation study. Patients receive alisertib orally (PO) twice daily (BID) on days 1-7 (dose levels 1-4) or days 1-3, 8-10, and 15-17 (dose levels 5-8). Patients also receive romidepsin intravenously (IV) over 4 hours on days 1 and 8 (dose levels 1-4) or 2, 9, and 16 (dose levels 5-8). Treatment repeats every 21 days (dose levels 1-4) or 28 days (dose levels 5-8) for up to 8 courses in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up for 4 weeks.