  Let W: R to (0,1] be continuous. Bernstein's approximation problem, posed in
1924, deals with approximation by polynomials in the weighted uniform norm
||fW|| Linfinity(R) . The qualitative form of this problem was solved by
Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the
problem were actively investigated starting from the 1960's. We survey old and
recent aspects of this topic, including the Bernstein problem, weighted Jackson
and Bernstein Theorems, Markov-Bernstein and Nikolskii inequalities, orthogonal
expansions and Lagrange interpolation. We present the main ideas used in many
of the proofs, and different techniques of proof, though not the full proofs.
The class of weights we consider is typically even, and supported on the whole
real line, so we exclude Laguerre type weights on [0,infinity). Nor do we
discuss Saff's weighted approximation problem, nor the asymptotics of
orthogonal polynomials.
