In this course we define both general and integral currents and cover all the foundational theorems and techniques pioneered by Federer and Fleming: the compactness theorem, the boundary rectifiability theorem, the costancy theorem, the deformation lemma, the slicing, and many others. We also highlight the special role played by sets of finite perimeter and functions of bounded variation and, if time allows, we cover the extension of the theory to Z/pZ coefficients.