A review is made of developments in the theory of the asymptotic solution of ordinary, linear, second-order differential equations with respect to a large parameter. It is shown that the theory is still not in a form where it can be applied rigorously to some problems in physics, and a new existence theorem which would overcome this difficulty is conjectured. This concerns the asymptotic equivalence of solutions of equations having the same turning points (and possibly singularities) arbitrary in number, in a given region of the real axis.