The composition operator is a well-known linear operator acting boundedly on several analytic function spaces. There is a profusion of remarkable results on the composition operator on the Hardy space Hp and the Bergman space Ap of the unit disk D, for instance. Less is known on domains with less regular boundary. In this talk, we give an overview of some results on less smooth domains. We will give a new boundedness result on Carleson domains, known as domains Ω (bounded or unbounded) such that for any Carleson measure μ, the L1(Ω, dμ)-norm of a function in H1 is less than C(μ)‖f ‖L1(∂Ω). We will provide some simple general domains for which the composition operator is bounded. It is based on a current joint work with B.R
