Construction and Analysis -- Transformation Analysis -- System Identification with Generalized Orthonormal Basis Functions -- Variance Error, Reproducing Kernels, and Orthonormal Bases -- Numerical Conditioning -- Model Uncertainty Bounding -- Frequency-domain Identification in ?2 -- Frequency-domain Identification in ?? -- Design Issues -- Pole Selection in GOBF Models -- Transformation Theory -- Realization Theory.Models of dynamical systems are of great importance in almost all fields of science and engineering and specifically in control, signal processing and information science. A model is always only an approximation of a real phenomenon so that having an approximation theory which allows for the analysis of model quality is a substantial concern. The use of rational orthogonal basis functions to represent dynamical systems and stochastic signals can provide such a theory and underpin advanced analysis and efficient modelling. It also has the potential to extend beyond these areas to deal with many problems in circuit theory, telecommunications, systems, control theory and signal processing. Nine international experts have contributed to this work to produce thirteen chapters that can be read independently or as a comprehensive whole with a logical line of reasoning: • Construction and analysis of generalized orthogonal basis function model structure; • System Identification in a time domain setting and related issues of variance, numerics, and uncertainty bounding; • System identification in the frequency domain; • Design issues and optimal basis selection; • Transformation and realization theory. Modelling and Identification with Rational Orthogonal Basis Functions affords a self-contained description of the development of the field over the last 15 years, furnishing researchers and practising engineers working with dynamical systems and stochastic processes with a standard reference work.