adaptive mesh refinement (amr) algorithms are one of two necessary tools for grand challenging problems in scientific computing. reliability of computer simulations is responsible for accurate computer predictions/designs. efficient and reliable a posteriori error estimation are, respectively, the key for success of amr algorithms and the reliability of computer predictions/designs. since babuska's pioneering work in 1976, the a posteriori error estimation has been extensively studied, and impressive progress has been made during the past four decades. however, due to its extreme difficulty, this important research field of computational science and engineering remains wide open. in this talk, i will describe (1) basic principles of the a posteriori error estimation techniques for finite element approximations to partial differential equations and (2) our recent work.