The theory of minimal surfaces in general Euclidean dimensions was developed in the 60's by De Giorgi, Reifenberg, Federer, Fleming, Almgren using measure theoretical methods. The approach due to De Giorgi is to interpret surfaces as boundaries of measurable sets E, and view the surface area as the perimeter of E which is defined as the BV norm of its characteristic function. A decade ago, we introduced with Caffarelli and Roquejoffre a nonlocal version of the perimeter functional which is relevant in the theory of phase-transitions with long-range interactions. In my lecture I will give an overview of the theory of the nonlocal minimal surfaces and discuss some of the more recent developments.