Multilingual dependency parsing encapsulates any attempt to parse multiple languages. It can involve parsing multiple languages in isolation (poly-monolingual), leveraging training data from multiple languages to process any of the included languages (polyglot), or training on one or multiple languages to process a low-resource language with no training data (zero-shot). In this thesis, we explore multilingual dependency parsing across all three paradigms, first analysing whether polyglot training on a number of source languages is beneficial for processing a target language in a zero-shot cross-lingual dependency parsing experiment using annotation projection. The results of this experiment show that polyglot training produces an overall trend of better results on the target language but a highly-related single source language can still be better for transfer. We then look at the role of pretrained language models in processing a moderately low-resource language in Irish. Here, we develop our own monolingual Irish BERT model gaBERT from scratch and compare it to a number of multilingual baselines, showing that developing a monolingual language model for Irish is worthwhile. We then turn to the topic of parsing Enhanced Universal Dependencies (EUD) Graphs, which are an extension of basic Universal Dependencies trees, where we describe the DCU-EPFL submission to the 2021 IWPT shared task on EUD parsing. Here, we developed a multitask model to jointly learn the tasks of basic dependency parsing and EUD graph parsing, showing improvements over a single-task basic dependency parser. Lastly, we revisit the topic of polyglot parsing and investigate whether multiview learning can be applied to the problem of multilingual dependency parsing. Here, we learn different views based on the dataset source. We show that multiview learning can be used to train parsers with multiple datasets, showing a general improvement over single-view baselines.