Commonsense reasoning research has so far been limited to English. We aim to evaluate and improve popular multilingual language models (ML-LMs) to help advance commonsense reasoning (CSR) beyond English. We collect the Mickey corpus, consisting of 561k sentences in 11 different languages, which can be used for analyzing and improving ML-LMs. We propose Mickey Probe, a language-general probing task for fairly evaluating the common sense of popular ML-LMs across different languages. In addition, we also create two new datasets, X-CSQA and X-CODAH, by translating their English versions to 14 other languages, so that we can evaluate popular ML-LMs for cross-lingual commonsense reasoning. To improve the performance beyond English, we propose a simple yet effective method — multilingual contrastive pretraining (MCP). It significantly enhances sentence representations, yielding a large performance gain on both benchmarks (e.g., +2.7% accuracy for X-CSQA over XLM-R_L).