We prove pointwise a posteriori error estimates for semi- and fully-discrete finite element methods for approximating the solution $u$ to a parabolic model problem.  Our estimates may be used to bound $\|u-u_h\|_{L_\infty(D)}$, where $D$ is an arbitrary subset of the space-time domain of definition of the given PDE.  When implemented in an adaptive method, these estimates should allow for efficient and accurate computation of $u$ on $D$ by requiring only enough mesh refinement away from $D$ in order to ensure that local solution quality is not polluted by global effects.