Braverman and Kazhdan proposed a general framework on the existence of Schwartz spaces, Fourier transforms, and summation formulae on reductive monoids that generalized Godement and Jacquet’s work. Their proposal was later refined by Ngô and extended to the setting of affine spherical varieties by Sakellaridis. In this talk, I will describe precise formulae for Fourier transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups, and certain spherical varieties built out of triples of quadratic spaces. This is based on the joint work with Jayce Getz and Spencer Leslie.