The aim of this research monograph is to establish strong, or Szeg? type asymptotics for extremal polynomials associated with weights W(x) := exp (-Q(x)) on . While the Q(x) treated are fairly general - even and of smooth polynomial growth at infinity - a typical example is Q(x) := , > 0. The results are consequences of a strengthened form of the following assertion: Given 0 > 1. Auxiliary results include inequalities for weighted polynomials, and zeros of extremal polynomials. The monograph is fairly self-contained, with proofs involving elementary complex analysis, and the theory of orthogonal and extremal polynomials. It should be of interest to research workers in approximation theory and orthogonal polynomials.