Isometric embedding of two dimensional Riemannian manifolds

A two dimensional Riemannian manifold is an abstract surface sitting nowhere in particular, but which somehow has the structures imposed on it that a surface gets by sitting in Euclidean space, such as tangent spaces, a metric etc.

The question is whether such an abstract surface can be realized as a surface in three dimensional Euclidean space. This problem can be reduced to solving a certain Monge-Ampere equation.