Participants in this research study have been diagnosed with Peripheral T-Cell Lymphoma (PTCL), these individuals did not respond to at least one prior treatment or their treatment is no longer effective. The purpose of the study is to assess the anti-cancer activity of alisertib as measured by response rate in patients with relapsed or refractory PTCL (i.e. the number of patients who have improvement in their relapsed or refractory PTCL (relapsed cancers that have come back after treatment, and refractory cancers that have not been successfully treated with prior therapy)) as compared to other PTCL treatments. The purpose is to also assess additional measures of anti-cancer activity, including how long the cancer is under control (if this occurs) and how long until the disease is no longer under control (otherwise known as disease progression) for patients on alisertib as compared to patients administered the other drugs used to treat PTCL included in this study. The study may have as many as 354 participants worldwide.