Abstract: One of the fundamental problems in Riemannian geometry is the existence of isometric immersions from one manifold into another, notably resulting in Nash's embedding theorem, and closely tied to general relativity, such as in Choquet-Bruhat's solution to the Cauchy problem. The Gauss, Ricci and Codazzi equations are well-known as the structure equations, meaning that any submanifold of any Riemannian manifold must satisfy them. A classical result (the fundamental theorem of submanifold theory) states that, conversely, they are sufficient conditions for a (semi-)Riemannian n-manifold to admit an immersion in the Euclidean space R^m. Proving fundamental theorems if the ambient space is not of constant sectional curvature is technically very difficult and there are only few other results known. I will discuss fundamental theorems for immersions into new classes of ambient manifolds, namely homogeneous spaces and special warped products, and discuss applications, such as non-existence results of associated families of minimal surfaces, existence of sister surfaces, and a new approach to the Cauchy problem of general relativity in the non-vacuum case.