In order to simulate human language capacity, natural language processing systems must complement the explicit information derived from raw text with the ability to reason about the possible causes and outcomes of everyday situations. Moreover, the acquired world knowledge should generalise to new languages, modulo cultural differences. Advances in machine commonsense reasoning and cross-lingual transfer depend on the availability of challenging evaluation benchmarks. Motivated by both demands, we introduce Cross-lingual Choice of Plausible Alternatives (XCOPA), a typologically diverse multilingual dataset for causal commonsense reasoning in 11 languages. We benchmark a range of state-of-the-art models on this novel dataset, revealing that current methods based on multilingual pretraining and zero-shot fine-tuning transfer suffer from the curse of multilinguality and fall short of performance in monolingual settings by a large margin. Finally, we propose ways to adapt these models to out-of-sample resource-lean languages where only a small corpus or a bilingual dictionary is available, and report substantial improvements over the random baseline. XCOPA is available at github.com/cambridgeltl/xcopa.