This open-label, multicenter, global study is designed to determine the recommended phase 2 dose, safety, efficacy, and pharmacokinetics/pharmacodynamics of durvalumab in subjects with certain lymphoma subtypes or CLL. Globally, 265 subjects may be enrolled into 4 treatment arms, including durvalumab monotherapy; durvalumab in combination with lenalidomide± rituximab; ibrutinib; or rituximab ± bendamustine. The study will have 3 parts: dose finding, dose confirmation, and dose expansion. Subjects receiving monotherapy may receive combination therapy or involved-field radiation to a single nodal site at time of progressive disease.