Using the “finite horizon condition”, we create surfaces embedded in three-dimensional Euclidean space for which the geodesic flow is ergodic with positive Lyapunov Exponent almost everywhere. These examples are not stably ergodic – under a small perturbation, they can become non-ergodic. By modifying the construction, we are able to create embedded surfaces with Anosov geodesicflow; these are the first examples of embedded surfaces with stably ergodic geodesic flow.