These are the lecture notes of the ESSLLI 2009 second week course on Abstract Categorial Grammar. This was an advanced course, while an introductory course was given the first week: Introduction to Abstract Categorial Grammars: Foundations and main properties, delivered by Philippe de Groote and Sylvain Salvati. An up-to-date version of these notes, possible errata, and generally ACG papers, can be found at the ACG homepage: http://calligramme.loria.fr/acg. At this URL, the ACG Development Toolkit is also available and downloadable as free software. It might be useful to run some of the examples given in these notes. In particular, some of the latter are gathered in a special file. The Abstract Categorial Grammar (ACG) framework, a grammar formalism based on the typed lambda calculus, elegantly generalizes and unifies a variety of grammar formalisms that have been proposed for the description of formal and natural languages. The first part of the course investigated formal-language-theoretic properties of "second-order" ACGs, a subclass of ACGs that have "context-free" derivations. Their generative capacity was precisely characterized. Lecture notes for this part are available at http://research.nii.ac.jp/~kanazawa/publications/esslli2009_lectures.pdf. An efficient Earley-style algorithm, suitable both for parsing and generation, was then presented. The second part of the course (these present notes) presents linguistic applications of ACGs and gives various illustrations of how ACGs provide flexible and explicit ways to model the syntax-semantics interface of natural language.