A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a preset tolerance. Again, numerical results demonstrate the efficiency of our approach. (C) 1998 Elsevier Science S.A. All rights reserved.