The trial is an open-label, dose-escalation trial to determine the safety and tolerability of Anti-KIR (1-7F9) in subjects with relapsed or refractory multiple myeloma (RRMM). A 3+3 design will be employed for the first dosing cycle at each dose level. The 7 planned dose levels are 0.0003 mg/kg, 0.003 mg/kg, 0.015 mg/kg, 0.075 mg/kg, 0.3 mg/kg, 1.0 mg/kg and 3.0 mg/kg. The subjects will receive up to a total of 4 administrations of Anti-KIR (1-7F9) with a dosing interval between each administration of 4 weeks. Safety, toxicity, PK (pharmacokinetic) and PD (pharmacodynamic) obtained in the first 4 weeks after dosing per group will be the basis for dose-escalation decisions. There will be follow-up visits every week the one month after the first administration and every two weeks following the second, third and fourth administrations. After the last administration there will be follow-up visits every month until KIR occupancy is no longer detected.