This is an open-label (all people know the identity of the intervention), multicenter (study conducted at multiple sites), dose escalation study. The study consists of 3 phases, including, the screening phase (within 14 days prior to the first study medication), the treatment phase, and the follow up phase. In the treatment phase, patients with recurrent mature B-cell neoplasms will be divided into 2 cohorts: Cohort 1 (consisting of between 3 and 12 patients), and Cohort 2 (consisting of between 6 and 12 patients). Cohort 1 will be further divided into 2 phases: a single dose (SD) phase and a multiple dose (MD) phase. During the initial SD phase, patients will first receive a single dose of PCI-32765 at 140 mg. After a washout period (period when the participant is not receiving any study medication) of between 72 and 168 hours, patients will then receive a second single dose of PCI-32765 at 280 mg. Following a second washout period, patients will enter the MD phase, where they will receive PCI-32765 at multiple doses of 420 mg per day for 35 days during the first cycle (35 days in Cycle 1) and for 28 days during the second cycle (28 days in Cycle 2) and every cycle thereafter. In cohort 2, patients will receive multiple doses of 560 mg per day for 35 days in Cycle 1, 28 days in Cycle 2, and every cycle thereafter. The patient's registration in Cohort 2 will be started after tolerability of Cohort 1 is confirmed. Tolerability of each dose level will be evaluated based on the dose-limiting toxicity (DLT) occurrence rate in Cycle 1 of each Cohort. Following the tolerability of the 420 mg/day dose level in subjects with mature B-cell neoplasms is confirmed in Cohort 1, a CLL/SLL Cohort will be added to further evaluate the safety and tolerability of this dose in this specific population in Japanese since 420 mg/day is the globally recommended dose for subjects with CLL and SLL, which are specific disease entities within mature B-cell neoplasms. Between 6 and 12 subjects will be enrolled in the CLL/SLL Cohort and will receive continuous dosing of PCI-32765 at 420 mg/day for 35 days in Cycle 1 and for 28 days in Cycle 2 and thereafter. Safety evaluations for adverse events, clinical laboratory tests, electrocardiogram, vital signs, Eastern Cooperative Oncology Group (ECOG) performance status, physical examination, and corneal eye examination will be monitored throughout the study.