The contents of this monograph fall within the general area of nonlinear functional analysis and applications. We focus on an important topic within this area: On invariant approximation and fixed points, a topic of intensive research efforts. Most of the results in Approximation Theory and Fixed Point Theory are available in Hilbert spaces and normed linear spaces and the consideration of approximation and fixed point problems in more general spaces viz. convex metric linear spaces, metric linear spaces, convex metric spaces and metric spaces is quite challenging. Since the results available in these more general spaces do not constitute a unified theory, we make an attempt in this direction. We assume familiarity with basic concepts of analysis and topology. The monograph is addressed to graduate students of mathematics, computer science, statistics, informatics, engineering, to mathematicians interested in learning about the subject, and to numerous specialists in the area.