This Phase 1/2, multi-center, open-label, multiple-dose, dose escalation study will evaluate the combination of elotuzumab and bortezomib in subjects with MM following 1 to 3 prior therapies. For the Phase 1 portion, elotuzumab will be administered by intravenous (IV) infusion at up to 4 dose levels ranging from 2.5 mg/kg to 20.0 mg/kg within 30 minutes following the administration of bortezomib at 1.3 mg/m^2 IV bolus. Bortezomib will be given in 21 day cycles (twice weekly for 2 weeks on Days 1, 4, 8, and 11 followed by a 10-day rest period). Elotuzumab will be administered as a separate infusion within 30 minutes following bortezomib administration on the same days as the first and last dose of each bortezomib cycle (i.e., Days 1 and 11).