The fractional p-Laplacian is a nonlinear, nonlocal operator with fractional order and homogeneity exponent p>1, arising in game theory and extending (in some sense) both the classical p-Laplacian and the linear fractional Laplacian. While behaving similarly to its local counterpart from the point of view of variational and topological methods, this operator requires an "ad hoc" approach in regularity theory. We will give an account on some regularity results for elliptic equations driven the fractional p-Laplacian, either free or coupled with nonlocal Dirichlet conditions: in particular we will discuss interior and boundary Hölder continuity, a special form of weighted Hölder regularity, and a recent local clustering lemma. Finally, we will rapidly hint at some applications such as comparison principles, Hopf type lemmas, Harnack inequalities, and an equivalence principle between Sobolev and Hölder minimizers of the associated energy functional. The talk is mainly based on some very recent collaborations with F.G. Düzgün, S. Mosconi, and V. Vespri.