While emotions are universal aspects of human psychology, they are expressed differently across different languages and cultures. We introduce a new data set of over 530k anonymized public Facebook posts across 18 languages, labeled with five different emotions. Using multilingual BERT embeddings, we show that emotions can be reliably inferred both within and across languages. Zero-shot learning produces promising results for low-resource languages. Following established theories of basic emotions, we provide a detailed analysis of the possibilities and limits of cross-lingual emotion classification. We find that structural and typological similarity between languages facilitates cross-lingual learning, as well as linguistic diversity of training data. Our results suggest that there are commonalities underlying the expression of emotion in different languages. We publicly release the anonymized data for future research.