The Bergman kernel is one of the important tools in several complex variables. In general, it is hard to compute the explicit form of the Bergman kernel for a given bounded domain. In this talk, we explain the computation of the Bergman kernel for some complex ellipsoids. It is expressed in terms of hypergeometric functions which are solutions of some second-order differential equations. We explain the boundary behavior of the Bergman kernel function using the theory of hypergeometric functions. We also explain the recent results on the intersection of cylinder-type domains.