Background: As lenalidomide becomes increasingly established for upfront treatment of multiple myeloma, patients refractory to this drug represent a population with an unmet need. The combination of pomalidomide, bortezomib, and dexamethasone has shown promising results in phase 1/2 trials of patients with relapsed or refractory multiple myeloma. We aimed to assess the efficacy and safety of this triplet regimen in patients with relapsed or refractory multiple myeloma who previously received lenalidomide. Methods: We did a randomised, open-label, phase 3 trial at 133 hospitals and research centres in 21 countries. We enrolled patients (aged ≥18 years)with a diagnosis of multiple myeloma and measurable disease, an Eastern Cooperative Oncology Group performance status of 0–2, who received one to three previous regimens, including a lenalidomide-containing regimen for at least two consecutive cycles. We randomly assigned patients (1:1)to bortezomib and dexamethasone with or without pomalidomide using a permutated blocked design in blocks of four, stratified according to age, number of previous regimens, and concentration of β2 microglobulin at screening. Bortezomib (1·3 mg/m2)was administered intravenously until protocol amendment 1 then either intravenously or subcutaneously on days 1, 4, 8, and 11 for the first eight cycles and subsequently on days 1 and 8. Dexamethasone (20 mg [10 mg if age >75 years])was administered orally on the same days as bortezomib and the day after. Patients allocated pomalidomide received 4 mg orally on days 1–14. Treatment cycles were every 21 days. The primary endpoint was progression-free survival in the intention-to-treat population, as assessed by an independent review committee. Safety was assessed in all patients who received at least one dose of study medication. This trial is registered at ClinicalTrials.gov, number NCT01734928; patients are no longer being enrolled. Findings: Between Jan 7, 2013, and May 15, 2017, 559 patients were enrolled. 281 patients were assigned pomalidomide, bortezomib, and dexamethasone and 278 were allocated bortezomib and dexamethasone. Median follow-up was 15·9 months (IQR 9·9–21·7). Pomalidomide, bortezomib, and dexamethasone significantly improved progression-free survival compared with bortezomib and dexamethasone (median 11·20 months [95% CI 9·66–13·73]vs 7·10 months [5·88–8·48]; hazard ratio 0·61, 95% CI 0·49–0·77; p<0·0001). 278 patients received at least one dose of pomalidomide, bortezomib, and dexamethasone and 270 patients received at least one dose of bortezomib and dexamethasone, and these patients were included in safety assessments. The most common grade 3 or 4 treatment-emergent adverse events were neutropenia (116 [42%]of 278 patients vs 23 [9%]of 270 patients; nine [3%]vs no patients had febrile neutropenia), infections (86 [31%]vs 48 [18%]), and thrombocytopenia (76 [27%]vs 79 [29%]). Serious adverse events were reported in 159 (57%)of 278 patients versus 114 (42%)of 270 patients. Eight deaths were related to treatment; six (2%)were recorded in patients who received pomalidomide, bortezomib, and dexamethasone (pneumonia [n=2], unknown cause [n=2], cardiac arrest [n=1], cardiorespiratory arrest [n=1])and two (1%)were reported in patients who received bortezomib and dexamethasone (pneumonia [n=1], hepatic encephalopathy [n=1]). Interpretation: Patients with relapsed or refractory multiple myeloma who previously received lenalidomide had significantly improved progression-free survival when treated with pomalidomide, bortezomib, and dexamethasone compared with bortezomib and dexamethasone. Adverse events accorded with the individual profiles of pomalidomide, bortezomib, and dexamethasone. This study supports use of pomalidomide, bortezomib, and dexamethasone as a treatment option in patients with relapsed or refractory multiple myeloma who previously received lenalidomide. Funding: Celgene.