  This is a survey article on an old topic in classical analysis. We present
some new developments in asymptotics in the last fifty years. We start with the
classical method of Darboux and its generalizations, including an uniformity
treatment which has a direct application to the Heisenberg polynomials. We then
present the development of an asymptotic theory for difference equations, which
is a major advancement since the work of Birkhoff and Trjitzinsky in 1933. A
new method was introduced into this field in the nineteen nineties, which is
now known as the nonlinear steepest descent method or the Riemann-Hilbert
approach. The advantage of this method is that it can be applied to orthogonal
polynomials which do not satisfy any differential or difference equations
neither do they have any integral representations. As an example, we mention
the case of orthogonal polynomials with respect to the Freud weight. Finally,
we show how the Wiener-Hopf technique can be used to derive asymptotic
expansions for the solutions of an integral equation on a half line.
