Two Riemannian manifolds are said to be isospectral if the associated Laplace-Beltrami operators have the same spectrum. Riemannian nilmanifolds have provided a rich source of examples of isospectral manifolds, exhibiting a wide variety of different phenomena. In particular, there exist continuous families of isospectral, nonisometric nil-manifolds, isospectral nilmanifolds for which the Laplacians acting on one-forms are not isospectral, and isospectral nilmanifolds that are not even locally isometric. This article reviews three different methods for constructing isospectral nilmanifolds and examines the geometry of resulting examples.