The aim of this talk is to describe several methods for numerically approximating the integral of a multivariate highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a method developed by Levin as a point of departure we will construct a new method that uses the same information as a Filon-type method, and obtains the same asymptotic order, while not requiring moments. This allows us to integrate over nonsimplicial domains, and with complicated oscillators. We also present a method for approximating oscillatory integrals with stationary points.