  We survey the main results of approximation theory for adaptive piecewise
polynomial functions. In such methods, the partition on which the piecewise
polynomial approximation is defined is not fixed in advance, but adapted to the
given function f which is approximated. We focus our discussion on (i) the
properties that describe an optimal partition for f, (ii) the smoothness
properties of f that govern the rate of convergence of the approximation in the
Lp-norms, and (iii) fast refinement algorithms that generate near optimal
partitions. While these results constitute a fairly established theory in the
univariate case and in the multivariate case when dealing with elements of
isotropic shape, the approximation theory for adaptive and anisotropic elements
is still building up. We put a particular emphasis on some recent results
obtained in this direction.
