During this talk I shall be mainly discussing some of the post processing methods for the gradient of the finite element approximation that render a superconvergent approximation to the gradient of the weak solution for a variety of problems. I shall follow the historical development of this field and emphasize how the evolution of this field has been influenced by the need for a posteriori error estmates that are based on superconvergent gradient recovery. Today, the selling point for gradient recovery techniques is no longer based on obtaining a superconvergent gradient in order to substitute the gradient of finite element approximation. In fact the focus of research has been on the use of the recovered gradients to provide a posteriori error estimation. In this context I shall discuss two types of estimators where superconvergent gradients can be employed, namely,

1. the Zienkiewicz-Zhu type error estimators and

2. estimators based on interpolation error bounds. Because of the superconvergence property, these estimators can be proved to be asymptotically exact.

This talk will follow the natural course of development in superconvergence theory. I shall start with some earlier results in this field and demonstrate how the subsequent results were mostly dedicated to addressing undesirable assumptions required by the former results. Examples include the high regularity of the weak solution, the simplicity of the problem and the regularity of the mesh. I shall focus primarily on how the regularity requirements on the mesh were relaxed as this seems essential under adaptive refinement of the finite element meshes. I shall finish my talk by mentioning some newer results in this field and current research in which I am engaged.