Let a real reductive algebraic group G act on a smooth real algebraic variety X. In this expository chapter we describe some recent qualitative results and conjectures on the multiplicities of irreducible representations \(\pi \) of G inside the space of Schwartz functions \(\mathcal {S} (X)\) . In particular, we give a sufficient condition for the finiteness of this multiplicity and a sufficient condition for this multiplicity to be zero, both in terms of the geometric relations between the associated variety of the annihilator of \(\pi \) and X. The results are stronger in the homogeneous case \(X=G/H\) , in which the multiplicity is equal to the dimension of the space of H-invariant functionals on the contragredient representation \(\widetilde {\pi }\) . This case also has applications to branching problems. We also describe p-adic analogs, which are mostly conjectural.