This paper studies the exact controllability and the stabilization of the cubic Schrodinger equation posed on a bounded interval. Both internal and boundary controls are considered, and the results are given first in a periodic setting, and next with Dirichlet (resp., Neumann) boundary conditions. It is shown that the systems with either an internal control or a boundary control are locally exactly controllable in the classical Sobolev space H(s) for any s >= 0. It is also shown that the systems with an internal stabilization are locally exponentially stabilizable in Hs for any s >= 0.