The phase I portion of the study is designed to determine the MTD of duvelisib with romidepsin and duvelisib with bortezomib. All patients must have a relapsed or refractory T-cell lymphoma. The design is a standard 3+3 dose escalation of two parallel phase I studies. Patients will be enrolled in Arms A (Romidepsin + duvelisib) and B (Bortezomib + duvelisib) of the study starting at dose level 1.