I will describe a one-parameter family of exact conservation laws for the Korteweg--de Vries equation. These permit us to obtain global uniform bounds on solutions for a wide range of Sobolev and Besov norms. In particular, we cover the cases $H^s$ for $-1\leq s < 1$. Our method treats the circle and the line cases on equal footing and applies also to other integrable systems such as (complex) mKdV and cubic NLS. This is joint work with M. Visan and X. Zhang.