A computer code for the method should be composed of two blocks each: solvers and checkers. In this book, we are mainly related to the problem (b) and try to present the main approaches developed for a posteriori error estimates in a variety of problems. The authors try to maintain a rigorous mathematical style, however, prove constructive if possible, and additional mathematical knowledge is presented when necessary. This book contains a number of new mathematical results and contains a list of methods posteriori error estimates which have been developed in a very recently.