Let G be a reductive group acting on a smooth affine variety X over a global field k. Consider the space of Schwartz functions on the adelic points X(A) of X, and let ξ be an element of X(k). Under what conditions does it make sense to define, purely by geometric means, a regularized orbital integral over the G(A)-orbit of ξ? This is a question that shows up on the geometric side of the trace formula, and its generalizations (such as the relative trace formula). I will present a new approach to this problem, that does not use truncation. If time permits, I will explain how the regularization of the orbital integral, whenever it can be defined, can be seen as a distribution on "the adelic points of the algebraic stack X/G".