Let H(U) be the set of all holomorphic functions on the unit ball U of the complex plane. The Hardy space H2 is the set of all functions f(z)= that belongs to H(U) such that . Let be a holomorphic self map of U. The composition operator C on H2 is defined as follows: C f fo , for all f H2 Littlewood's principle shows that C is a bounded operator on H2. Recall that, an operator T on a Hilbert space H is said to be cyclic operator if there exists a vector x in H, such that span {Tnx : n 0, 1, _} is dense in H, the operator T is supercyclic if there is a vector x in H, such that the set { nTnx : n , n 0, 1, _} is dense in H. It may happen that orb(T, x) {Tnx : n 0, 1, _} is dense in H, in this case T is called a hypercyclic operator. One of our main concerns in this thesis was to give conditions that are necessary and (or) sufficient for the composition operator to be a cyclic (hypercyclic, supercyclic) operator.We give some known results with details of the proofs, specially when is a linear fractional transformation, as mentioned in the original abstract This thesis contains some new results (to the best of our knowledge) for the cyclicity of the operator , where is the adjoint of the composition operator C .