Rigidity of Lagrangian intersections play a fundamental role in symplectic geometry and topology. In this talk, after a general introduction, I will address a natural Lagrangian intersection problem in the context of toric symplectic manifolds: displaceability of torus orbits. The emphasis will be on recent joint work with Leonardo Macarini on their non-displaceability and with Strom Borman and Dusa McDuff on their displaceability.