The text is focused on formulation, analysis, and computational procedures for reduced basis approximation and a posteriori error estimation for parametrized PDEs in the real-time and many-query contexts. The book also serves as background and (certain specific chapters) documentation for the companion rbMIT © MIT Software package. The book is intended to serve both "Developers" — numerical analysts and computational tool-builders — who wish to further develop the methodology, and "Users" — computational engineers and educators — who wish to apply the methodology to new applications. ("End Users" interested in the Worked Problems will not need the material in the book.) The book consists of General Preliminaries followed by nine Parts: each successive Part addresses a new class of problems in each case introduced by an appropriate abstraction and particular examples. We begin with parametrically coercive compliant problems and then proceed to general coercive elliptic problems (such as conduction, forced convection, and elasticity) and non-coercive elliptic problems (such as frequency-domain acoustics and elastodynamics); we next discuss parabolic problems (such as transient convection-diffusion); we then consider non-affine (in parameter) problems, and finally certain nonlinear problems (such as the incompressible Navier-Stokes equations).