A basic open problem in symplectic topology is to classify Lagrangian submanifolds (up to, say, Hamiltonian isotopy) in a given symplectic manifold.  In recent years, ideas from mirror symmetry have led to the realization that even the simplest symplectic manifolds (eg. vector spaces or complex projective spaces) contain many more Lagrangian tori than previously thought.  We will present some of the recent developments on this problem, and discuss some of the connections (established and conjectural) between Lagrangian tori, cluster mutations, and toric degenerations, that arise out of this story.