  In this article we study for the first time the regularity of the free
boundary in the one-phase free boundary problem driven by a general nonlocal
operator. Our main results establish that the free boundary is $C^{1,\alpha}$
near regular points, and that the set of regular free boundary points is open
and dense. Moreover, in 2D we classify all blow-up limits and prove that the
free boundary is $C^{1,\alpha}$ everywhere. The main technical tool of our
proof is an improvement of flatness scheme, which we establish in the general
framework of viscosity solutions, and which is of independent interest. All of
these results were only known for the fractional Laplacian, and are completely
new for general nonlocal operators. In contrast to previous works on the
fractional Laplacian, our method of proof is purely nonlocal in nature.
