Existing summarization datasets come with two main drawbacks: (1) They tend to focus on overly exposed domains, such as news articles or wiki-like texts, and (2) are primarily monolingual, with few multilingual datasets. In this work, we propose a novel dataset, called EUR-Lex-Sum, based on manually curated document summaries of legal acts from the European Union law platform (EUR-Lex). Documents and their respective summaries exist as cross-lingual paragraph-aligned data in several of the 24 official European languages, enabling access to various cross-lingual and lower-resourced summarization setups. We obtain up to 1,500 document/summary pairs per language, including a subset of 375 cross-lingually aligned legal acts with texts available in *all* 24 languages. In this work, the data acquisition process is detailed and key characteristics of the resource are compared to existing summarization resources. In particular, we illustrate challenging sub-problems and open questions on the dataset that could help the facilitation of future research in the direction of domain-specific cross-lingual summarization. Limited by the extreme length and language diversity of samples, we further conduct experiments with suitable extractive monolingual and cross-lingual baselines for future work. Code for the extraction as well as access to our data and baselines is available online at: [https://github.com/achouhan93/eur-lex-sum](https://github.com/achouhan93/eur-lex-sum).