The lake equations are introduced as a 2D model for the vertically averaged horizontal component of a 3D incompressible fluid. I will briefly present the model, then I will review the well-posedness results. In particular, I will often compare the results and technics with the 2D Euler equations. Next, I will discuss the stability of these equations when an island disappears by flooding, or at the opposite if an island appears (for instance, because of the decrease of the level water). This last work is in collaboration with Lars Eric Hientzsch and Evelyne Miot.