Parsing is a key task in natural language processing. It involves predicting, for each natural language sentence, an abstract representation of the grammatical entities in the sentence and the relations between these entities. This representation provides an interface to compositional semantics and to the notions of "who did what to whom." The last two decades have seen great advances in parsing English, leading to major leaps also in the performance of applications that use parsers as part of their backbone, such as systems for information extraction, sentiment analysis, text summarization, and machine translation. Attempts to replicate the success of parsing English for other languages have often yielded unsatisfactory results. In particular, parsing languages with complex word structure and flexible word order has been shown to require non-trivial adaptation. This special issue reports on methods that successfully address the challenges involved in parsing a range of morphologically rich languages (MRLs). This introduction characterizes MRLs, describes the challenges in parsing MRLs, and outlines the contributions of the articles in the special issue. These contributions present up-to-date research efforts that address parsing in varied, cross-lingual settings. They show that parsing MRLs addresses challenges that transcend particular representational and algorithmic choices.