The mirror symmetry conjecture (inspired by physics) has spurred a lot of development in symplectic geometry. In the last few years, a wave of modern homotopy theory has also entered the symplectic landscape, and begun to present new questions about the structure of symplectic manifolds. In this talk, we’ll explain a basic invariant in symplectic geometry (the Fukaya category) and, as time allows, give a survey of new inroads being opened through Lagrangian cobordisms, derived geometry, and deformation theory.