I will survey recent work on low regularity conservation laws, equicontinuity, and optimal well-posedness for completely integrable systems.  The centerpiece of my presentation is the method of commuting flows that we introduced to answer the well-posedness question in the non-perturbative regime of these equations. I will then describe the increasingly sophisticated techniques that have been required in order to achieve sharp results across a spectrum of integrable models.  This is based on joint work with B. Harrop-Griffiths, R. Killip, T. Laurens, and M. Ntekoume.