Given a riemmanian manifold (M,g) (of a given smoothness) is it possible to isometrically embed it (again with some smoothness) into a Euclidean space of sufficiently high dimension? This problem was solved affirmatively by Nash for the case that the metric tensor is continuous and the embedding is C^1 in 1954 and then for the case of C^r isometric embeddings of C^r Riemannian manifolds (r>3)in 1956. I will give some background to this problem and then focus on the solution to the C^1 case, attempting to highlight the stark difference between C^1 isometric embeddings and smoother ones. Finally, I will try to give some overview of the general analytical machinery, which replaced his more elementary geometric methods in the first paper, that Nash developed for the general case.