In this talk, I will review some regularity results for weak solutions of nonlocal variants of the p-Laplace equation. The model case is given by the Euler-Lagrange equation of an Aronszajn–Gagliardo–Slobodeckij seminorm. In particular, I will present a higher differentiability result for solutions, recently obtained in collaboration with Erik Lindgren (KTH). I will also discuss some connections of these equations to an Optimal Transport problem with congestion effects.