  We survey developments, over the last thirty years, in the theory of Shape
Preserving Approximation (SPA) by algebraic polynomials on a finite interval.
In this article, "shape" refers to (finitely many changes of) monotonicity,
convexity, or q-monotonicity of a function (for definition, see Section 4). It
is rather well known that it is possible to approximate a function by algebraic
polynomials that preserve its shape (i.e., the Weierstrass approximation
theorem is valid for SPA). At the same time, the degree of SPA is much worse
than the degree of best unconstrained approximation in some cases, and it is
"about the same" in others. Numerous results quantifying this difference in
degrees of SPA and unconstrained approximation have been obtained in recent
years, and the main purpose of this article is to provide a "bird's-eye view"
on this area, and discuss various approaches used.
  In particular, we present results on the validity and invalidity of uniform
and pointwise estimates in terms of various moduli of smoothness. We compare
various constrained and unconstrained approximation spaces as well as orders of
unconstrained and shape preserving approximation of particular functions, etc.
There are quite a few interesting phenomena and several open questions.
