In this chapter we study small perturbations of the KdV equation $$ u_t = - u_{xxx} + 6uu_x $$ on the real line with periodic boundary conditions. We consider this equation as an infinite dimensional, integrable Hamiltonian system and subject it to sufficiently small Hamiltonian perturbations. The aim is to show that large families of time-quasiperiodic solutions persist under such perturbations. This is true not only for this KdV equation, but in principle for all higher order KdV equations as well. As an example, the second equation in the KdV hierarchy will be considered in detail.