Brownian motion is the fundamental example of a stochastic process, that is, a real valued random function over the real line; what is its analogue if we replace the real line with some other manifold? Does any analogue exist? These questions were first considered by Paul Levy around 1950 for the case of Euclidean space and the sphere. Since then a number of researchers have looked at this question, considering also fractional Brownian motion, however until recently satisfactory extensions only existed for some special classes of manifolds and only certain fractional brownian motions. In this talk we will discuss a new approach whereby analogues of the full range of fractional Brownian motions are constructed over a wide class of manifolds. In doing so we will highlight a connection between fractional Brownian motion and the Laplacian that doesn't seem widely known (although we think it should be!). The tools we use come from spectral geometry, in particular we make essential use of the heat kernel and its estimates.