**Abstract**

This Chapter aims to investigate the error estimation of numerical approximation to a class of semilinear parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the finite element method for which the meshes are allowed to change in time. The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto 2003, enabling us to use the a posteriori error estimators derived for elliptic models and to obtain optimal order in *L*<sup>∞</sup> *H*<sup>1</sup> for Lipschitz and non-Lipschitz nonlinearities. In this Chapter, some challenges will be addressed to deal with nonlinear term by employing a continuation argument.

**Keywords:** A posteriori error estimates, semilinear parabolic problems, finite element approximation, *L*<sup>∞</sup> (*H*<sup>1</sup> ) bounds in finite element approximation, fully discrete semilinear parabolic approximation
