In the thesis, we study extensions of submanifold theory to settings which do not (necessarily) have the real numbers as ground field. An example of this is the study of submanifolds of holomorpic Riemannian manifolds. We show how holomorphic Riemannian submanifold theory has useful applications to a.o. the study of submanifolds for pseudo-Riemannian spaces. Furthermore, it will be shown in the thesis how Riemannian geometry and submanifold theory can be generalized using the concept of Lie-Rinehart algebras, and some practical applications will be given, resulting from this generalized theory.