In 1980, A. P. Calderon published a short seminal paper entitled "On an inverse boundary value problem", which has become the starting point in the mathematical analysis of the following inverse problem: Can one determine the electrical conductivity of a medium by making current and voltage measurements at the boundary of the medium? To this day, this problem serves as a fundamental source of motivation and inspiration for many developments in the field of inverse boundary problems. In this talk we shall give an introduction to the field of inverse boundary problems, survey some of the most important developments, and state some open problems.