Abstract:  The Bergman kernel function is a reproducing kernel for the space of square-integrable holomorphic functions in a domain. It plays an important role in several complex variables, since it is invariant under biholomorphic mappings. The explicit form of the Bergman kernel function differs depending on the domain. It is difficult to compute the explicit formula of the Bergman kernel function. The purpose of the talk is to explain the methods for computing the Bergman kernel function in various domains including unit ball, complex ellipsoids, Hartogs domain. We will also explain the existence of zeros of the Bergman kernel function, so called the Lu Qi-Keng problem.