Physicists have observed in the '80s that Calabi-Yau manifolds come in pairs so that quantum cohomology on the one is related to period integrals on the other. This phenomenon, known as mirror symmetry, has since evolved into a deeper understanding that symplectic geometry on a manifold is typically encoded in the complex geometry of another, its mirror. I will discuss in some simple examples how the relation arises naturally from the study of Hamiltonian Floer cohomology associated to invariant sets of an integrable system. I will start by discussing Floer cohomology on the cotangent bundle of the n-torus and its relation to rigid analytic geometry. Then I will discuss Floer cohomology of a number of singular torus fibrations, the nodal singularity in dimension two which is a self mirror, and, as time permits, the positive and negative singularities in dimension 3, which are a mirror to one another. A little more speculatively, I will try to explain how the phenomena of mirror symmetry can be understood as arising by patching together such local models. No knowledge of Floer homology or quantum cohomology will be assumed.