  A method for computing the Riesz $\alpha$-capacity, $0 < \alpha \le 2$, of a
general set $K \subset \mathbb{R}^d$ is given. The method is based on
simulations of isotropic $\alpha$-stable motion paths in $d$-dimensions. The
familiar Walk-On-Spheres method, often utilized for simulating Brownian motion,
is modified to a novel Walk-In-Out-Balls method adapted for modeling the stable
path process on the exterior of regions ``probed'' by this type of generalized
random walk. It accounts for the propensity of this class of random walk to
jump through boundaries because of the path discontinuity. This method allows
for the computationally efficient simulation of hitting locations of stable
paths launched from the exterior of probed sets. Reliable methods of computing
capacity from these locations are given, along with non-standard confidence
intervals. Illustrative calculations are performed for representative types of
sets K, where both $\alpha$ and $d$ are varied.
