This report describes the parsing problem for Combinatory Categorial Grammar (CCG), showing how a combination of Transformer-based neural models and a symbolic CCG grammar can lead to substantial gains over existing approaches. The report also documents a 20-year research program, showing how NLP methods have evolved over this time. The staggering accuracy improvements provided by neural models for CCG parsing can be seen as a reflection of the improvements seen in NLP more generally. The report provides a minimal introduction to CCG and CCG parsing, with many pointers to the relevant literature. It then describes the CCG supertagging problem, and some recent work from Tian et al. (2020) which applies Transformer-based models to supertagging with great effect. I use this existing model to develop a CCG multitagger, which can serve as a front-end to an existing CCG parser. Simply using this new multitagger provides substantial gains in parsing accuracy. I then show how a Transformer-based model from the parsing literature can be combined with the grammar-based CCG parser, setting a new state-of-the-art for the CCGbank parsing task of almost 93% F-score for labelled dependencies, with complete sentence accuracies of over 50%.