All computational results obtained from computer simulations of
physical phenomena involve numerical error. Discretization error can
be large, pervasive, unpredictable by classical heuristic means, and
can invalidate numerical predictions. The {\it a posteriori} error
estimation is a mathematical theory for estimating and quantifying
discretization error based on information gained during the current
stage of the computation. In this talk, I will first give an
introductory review of existing {\it a posteriori} error estimators,
and then introduce two new estimators. One of them is a modification
of the recovery-based estimator and the other is exactly equal to
the true error in an ``energy'' norm on any given mesh. Numerical
examples will also be presented.