Tropical geometry around Lagrangian submanifolds

Given a closed symplectic manifold X, and a differentiable manifold L, can we embed L into X as a Lagrangian submanifold? This central question in Symplectic Geometry is far from being resolved. According to the Symplectic Field Theory approach (as proposed by Eliashberg, Givental and Hofer about 20 years ago) if L is embeddable to X then enumerative invariants of TL-L and X must be compatible. If L is a real torus then TL coincides with (C*)^n and its enumerative geometry is described by the tropical geometry in R^n. In this talk we'll look at some other examples of L when tropical geometry comes into play, including the famous "conifold transition" case of L=S^3 and its finite quotients. We'll also consider some cases when L is disconnected or singular.