In this short course we will present the isoperimetric problem in a Euclidean space with density. In other words, one wants to minimize, as usual, the perimeter of sets with a given volume; however, the "perimeter" and the "volume" of a set are given by the integral of two given functions (the densities) on the set and on its boundary respectively. The classical case clearly corresponds to the choice of both functions constantly equal to $1$.
This is a quite classical question, but there are also very recent developments. We will try to give a general overview of the subject, and to explain what is known and some of the most important open problems which are left. We will also describe some of the key ideas of the proofs, without entering too much into the technical details.
In the first of the three lessons we will discuss the problem, give an idea of the classical known results, and individuate the main questions that one wants to answer. Then, roughly speaking, the second lesson will be devoted to study the questions concerning the regularity of isoperimetric sets, and the last one the questions concerning the existence.