I will present some new results on the obstacle problem for the fractional Laplace operator with drift in the subcritical regime. The central one consists in the $C^{1+\gamma}$-H\"older regularity of the regular part of the free boundary. The method of the proof is based on a new monotonicity formula and an epiperimetric inequality. These results generalize those in G. Weiss’ 1999 Inventiones paper for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift.