We present CrossSum, a large-scale cross-lingual summarization dataset comprising 1.68 million article-summary samples in 1,500+ language pairs. We create CrossSum by aligning parallel articles written in different languages via cross-lingual retrieval from a multilingual abstractive summarization dataset and perform a controlled human evaluation to validate its quality. We propose a multistage data sampling algorithm to effectively train a cross-lingual summarization model capable of summarizing an article in any target language. We also introduce LaSE, an embedding-based metric for automatically evaluating model-generated summaries. LaSE is strongly correlated with ROUGE and, unlike ROUGE, can be reliably measured even in the absence of references in the target language. Performance on ROUGE and LaSE indicate that our proposed model consistently outperforms baseline models. To the best of our knowledge, CrossSum is the largest cross-lingual summarization dataset and the first ever that is not centered around English. We are releasing the dataset, training and evaluation scripts, and models to spur future research on cross-lingual summarization. The resources can be found at https://github.com/csebuetnlp/CrossSum