One of the most effective methods for solving boundary value problems for basic elliptic equations of mathematical physics in a given domain is the method of layer potentials. Its essence is to reduce the entire problem to an integral equation on the boundary of the domain which is then solved using Fredholm theory. Until now, this approach has been primarily used in connection with second order operators for which a sophisticated, far-reaching theory exists. This stands in sharp contrast with the case of higher order operators (arising for instance in plate elasticity) for which very little is known in this regard. In this talk I will survey recent results aimed at extending the method of singular integral operators (of layer potential type) to the higher order setting. This can be done in a very general class of domains which is in the nature of best possible from the point of view of geometric measure theory. The talk is intended for a general audience.