The authors propose a new point of view on graph connectivity that is based on geometric and physical intuition. The main theorem is a geometric characterization of k-vertex connected graphs. It says that a graph G is k-connected if and only if G has a certain 'nondegenerate convex embedding' in R**k- **1 . Probabilistic algorithms for computing the connectivity of a graph are given. The first is a Monte Carlo algorithm that runs in time O(n**2 **. **5 plus nk**2 **. **5 ), where n is the number of vertices and k is the vertex connectivity of the input graph. The second is a Las Vegas algorithm (i. e. , one that never errs) that runs in expected time O(kn**2 **. **5 plus nk**3 **. **5 ).