In this talk we prove that the first two members of the KdV hierarchy have the non-squeezing property, that is, they cannot map a ball in the symplectic space H^{-\frac{1}{2}} wholly inside a thinner cylinder. This is an infinite-dimensional analogue of Gromov's famous symplectic non-squeezing theorem. Our proof relies on an approximation scheme by finite-dimensional Hamiltonian systems that exploits the complete integrability of these equations.