The authors formulate and analyze Schwarz methods for solving biharmonic problems, where the biharmonic operator contains up to fourth-order derivatives. For the biharmonic problems, the Schwarz method requires two boundary conditions, but not just one as for the classical Laplace operator. This implies also that Schwarz methods will need to use two transmission conditions between subdomains. As shown by the authors in [SIAM J. Sci. Comput. 43, No. 3, A1881--A1906 (2021; Zbl 1477.65259)], there are many choices of Dirichlet and Neumann conditions for biharmonic problems, which leads to various Dirichlet-Neumann domain decomposition algorithms with different convergence rates. In this paper, the authors first study the classical Schwarz method by choosing different Dirichlet conditions, and show that the convergence rate for two of them is much better than the classical choice, such as for the Laplace problem. Then, the authors propose new optimized Schwarz methods by linearly combining different Dirichlet and Neumann conditions, and further show that, by optimizing the Robin matrices, the convergence rates become the same as for optimized Schwarz methods for the Laplace problem which is better than what is usually achieved for the biharmonic equation. Numerical examples are presented to demonstrate the theoretic analysis.