Spherical varieties form a wide and interesting class of almost homogeneous spaces which includes symmetric ones. They have relevance to the Langlands program, as they appear in the analysis of period integrals and L-functions, and in fact we now have conjectures describing their L^2-spectrum in terms of certain distinguished Arthur parameters. After explaining this motivation, I will describe several steps towards developing a Plancherel formula for spherical varieties of split groups over p-adic fields, which are in agreement with the conjectures. In most cases we obtain a Plancherel formula up to discrete spectra. This is joint work with Akshay Venkatesh, and uses in a crucial way an unpublished argument of Joseph Bernstein.