Despite its innocuous appearance, the 1d cubic NLS is a truly remarkable PDE. Not only does it arise as a model in numerous physical scenarios, for example fluid dynamics and nonlinear optics, but it is also part of the select group of integrable equations, in the sense that it possesses a Lax pair and infinitely many conserved quantities. Building on the work of Killip and Visan on the KdV equation, in this talk we present a proof of well-posedness for the cubic NLS that combines its deep mathematical structure with robust PDE techniques to obtain a sharp result in Sobolev spaces. We will also discuss the corresponding results for an intimately related equation, the mKdV. This is joint work with Rowan Killip and Monica Visan.