Here we survey some previously published results and announce some that have been newly obtained. We first review some of the results in  on estimates for the finite element error at a point. These estimates and analogous ones in  and  have been applied to problems in a posteriori estimates , , superconvergence  and others , . We then discuss the extension of these estimates to local estimates in L∞ based negative norms. These estimates have been newly obtained and are applied to the problem of obtaining an asymptotically exact a posteriori estimator for the maximum norm of the solution error on each element.