Recently, a number of formulas reminiscent of Weyl's law have been discovered in the context of symplectic geometry. Various three-manifold invariants, defined by building on ideas originating in Morse theory, have been at the heart of these developments. I will give a sense for how this works by surveying some of these symplectic Weyl laws. I will briefly explain an application to the generic density of periodic points, settling a special case of what is known as the Closing Lemma.