We present an overview of the theory and applications of spectral penalty methods for the stable solution of initial boundary value problems. Through a number of case studies we shall illustrate that imposing boundary conditions weakly through a penalty term rather than strongly as is classically done, removes many of the problems traditionally associated with the construction and analysis of stable and accurate pseudospectral approximations, The discussion includes the treatment of complex boundary operators, the construction of integration preconditioners and the formulation of spectral methods on general unstructured grids, highlighting the versatility and flexibility introduced by the elegant splitting of the operator and the boundary conditions exploited in the penalty methods, (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.