Modeling language at the syntactic or semantic level is a key problem in natural language processing, and involves a challenging set of structured prediction problems. In this talk I'll describe work on machine learning approaches for syntax and semantics, with a particular focus on lexicalized grammar formalisms such as dependency grammars, tree adjoining grammars, and categorial grammars. I'll address key issues in the following areas: 1) the design of learning algorithms for structured linguistic data; 2) the design of representations that are used within these learning algorithms; 3) the design of efficient approximate inference algorithms for lexicalized grammars, in cases where exact inference can be very expensive. In addition, I'll describe applications to machine translation, and natural language interfaces.