In this research report, we mainly study the boundedness, compactness, weak compactness and other properties of composition operators on some analytic function spaces. The report consists of four chapters.Chapter 1 is an introduction of this report.In Chapter 2, we give some necessary and sufficient conditions for the compactness of the product operators of a composition operator and another one's adjoint on Bergman space. These extend some previous results of a single composition operator on Bergman space. We also point out some potential generalization to vector-valued setting.Chapter 3 is devoted to some composition operators between different Nevanlinna type classes. Some characterizations of composition operators, which improve some integrability of function, are given. In addition, we also characterize the inducing maps which induce invertible or Fredholm composition operators on Nevanlinna type classes.In Chapter 4 some results in Chapter 3 are generalized to vector-valued setting. More precisely, we study composition operators on vector-valued analytic Nevanlinna algebras. In particular, we prove that the weak compactness of composition operators on vector-valued Nevanlinna algebras is equivalent to those of composition operators on vector-valued Hardy and weighted Bergman spaces, respectively.