Includes bibliographical references (pages 409-418) and indexes

Introduction -- Part 1. Basics. Background material -- Dynamical background -- Rotations -- Group extensions -- Part 2. Cubes. Cubes in an algebraic setting -- Dynamical cubes -- Cubes in ergodic theory -- The structure factors -- Part 3. Nilmanifolds and nilsystems. Nilmanifolds -- Nilsystems -- Cubic structures in nilmanifolds -- Factors of nilsystems -- Polynomials in nilmanifolds and nilsystems -- Arithmetic progressions in nilsystems -- Part 4. Structure theorems. The ergodic structure theorem -- Other structure theorems -- Relations between consecutive factors -- The structure theorem in a particular case -- The structure theorem in the general case -- Part 5. Applications. The method of characteristic factors -- Uniformity seminorms on [[infinity] -- Multiple correlations, good weights, and anti-uniformity -- Inverse results for uniformity seminorms and applications -- The comparison method