After a brief introduction to the theory of integral currents, I will focus on the regularity theory for area-minimizing ones. The aim is to prove the partial regularity theorem of Almgren, which shows that such objects are regular submanifolds except for a closed singular set of codimension at most two. In the course, I will follow my recent work with Emanuele Spadaro where we give a simpler approach to Almgren's result.