In this lecture we will review the Plancherel formula that we got by looking to neighborhoods at infinity of the symmetric spaces and then using the method of Sakellaridis-Venkatesh for spherical varieties for a split group. For us the group is not necessarily split.
We will try to present in the last part a joint work with Pascale Harinck and Yiannis Sakellaridis in which we prove Paley-Wiener theorems for symmetric spaces.