Let $\Omega$ be a region in $\mathbb C^d$. For $0<p<\infty$ the Bergman space $L^p_a(\Omega)$ consists of all analytic functions on $\Omega$ that are $p$-integrable with respect to Lebesgue measure on $\Omega$. The name honors Stefan Bergman, who was the first to systematically study the space $L^2_a(\Omega)$ and its reproducing kernel, and applied his results to questions in conformal mapping and boundary value problems. Nowadays the theory of Bergman spaces is comprised of many different areas in one and several complex variables.

In these talks we will discuss some aspects of the theory in the case when $\Omega$ is the open unit disc in the complex plane. In this case the Bergman spaces contain the classical Hardy spaces, and for many Hardy space theorems one may ask whether there that are analogous statements for the Bergman spaces. Some of the questions of this type have been answered, and some are still open. Sometimes the answers have led to entirely new questions.

Topics that we plan to touch upon will include Bergman projections and Bergman kernels, invariant subspaces, contractive divisors, and inner-outer factorizations.