Abstract: The problem of isometry-invariant geodesics, introduced by K. Grove in the 70s, is a generalization of the closed geodesics one: given an isometry of a closed Riemannian manifold, one looks for geodesics on which the isometry acts as a non-trivian translation. In this talk, after recalling the framework of the problem and possible symplectic generalizations, we present a few new multiplicity results on certain Riemannian manifolds non-trivially homeomorphic to a product, and on Riemannian manifolds with infinite abelian fundamental group.