Kernel density estimation for univariate long-memory processes is known to lead to a smoothing dichotomy. In the first part of this talk, consequences for optimal bandwidth choice are discussed. In particular, higher order kernels turn out to play an important role. In the second part, a multivariate reduction principle for the empirical process is used to obtain generalizations to multivariate long-memory processes. Estimation of multivariate derivatives is also discussed. This is joint work with Nadja Schumm and Klaus Telkmann.