There is great current interest in the use of diffusion maps for dimension reduction. We discuss some examples of diffusion methods applied to understanding dynamical data, in particular combining spectral approaches with delay coordinates. In addition, we extend the usual diffusion map construction by introducing local kernels, a generalization of the standard isotropic kernel. In fact, for data lying on a manifold, any Riemannian geometry can be generated by an appropriate local kernel. This work is joint with Tyrus Berry.