This thesis gives a comprehensive overview of the Basis Function Models (BMs), a family of computational models of expressive music performance. These models have been developed over the past years, and have been steadily growing in complexity. The motivation for this work is to model the complex relationship between properties and structure of a given composition, and musically plausible ways of playing the piece expressively and in this way also to learn more about this complex art. The focus is on Western classical music mostly on the piano, but also with recent extensions towards complex orchestral pieces. The basic idea in the BM framework is that structural properties of a musical piece (given as a score), which are believed to be relevant for performance decisions, can be modeled in a simple and uniform way, via so-called basis functions: numeric features that capture specific aspects of a musical note and its surroundings. A predictive model of performance can then predict appropriate patterns for expressive performance dimensions such as tempo, timing, dynamics, and articulation from these basis functions. A central methodological principle in this work is to take a data-driven approach: the model is not constructed manually, based on musical knowledge or hypotheses, but is learned from a large collection of real human performances, via state-of-the-art linear and non-linear machine learning algorithms. In this way, it is the empirical data that dictates what the model will look like, and an analysis of the learned models can provide interesting insights into the complex relation between score and performance. A series of BMs of growing complexity will be described. The models are evaluated on corpora of classical piano and symphonic music recordings, in terms of their ability to accurately predict a performers actual choices. In addition, some qualitative insights gained from an analysis of the models will be presented. Furthermore, recent developments towards integrating the basis function model into a reactive, real-time accompaniment system will be described. This work concludes with a critical and comprehensive survey of the current state-of-the-art in computational models of expressive performance.