Drought duration and severity are strongly dependent, so return periods estimated from each variable alone can misrepresent drought risk. This study built bivariate copula models of drought duration (D) and severity (S) at eleven sites on five continents, from arid Lahore to humid Abashiri, using the 6-month Standardized Precipitation Evapotranspiration Index (SPEI-6) computed from 108 years (1900–2007) of monthly climatic water balance. Drought events were identified by run theory (threshold −0.5, pooling of events separated by one month and removal of one-month events), which gave 50–67 events per site. Kendall’s τ between D and S was 0.64–0.78. The lognormal distribution fitted duration best at 10 sites, and severity was described by lognormal, gamma, Weibull or exponential distributions. Gaussian, Clayton, Gumbel and Frank copulas were fitted by maximum likelihood and tested with a Cramér–von Mises test adapted to integer durations. The Clayton copula was rejected at 6 sites. The Gumbel copula, whose upper-tail dependence coefficient (0.69–0.83) matched the empirical estimates, was selected at 10 sites. For events whose duration and severity both have 50-year univariate return periods, the joint "AND" return period was 60–76 years, the Kendall return period 55–63 years and the "OR" return period 37–43 years. Assuming independence overestimated the AND return period 16–27-fold (1189–1748 years). Parametric bootstrap showed that the 90% interval of the AND return period spanned a factor of 4–7. The largest observed droughts, including 2001–2004 at Albuquerque and 1997–2000 at Valencia, had Kendall return periods of 50–397 years. Copula choice changed the AND return period of the largest events by up to a factor of 43. More drought events occurred in 1954–2007 than in 1900–1953 at 10 of the 11 sites. Upper-tail-dependent copulas and explicit uncertainty bounds are recommended for drought risk design.