Optimization problems often involve vector norms, which has led to extensive research on developing algorithms that can handle objectives beyond the $\ell_p$ norms. Our work introduces the concept of submodular norms, which are a versatile type of norms that possess marginal properties similar to submodular set functions. We show that submodular norms can accurately represent or approximate well-known classes of norms, such as $\ell_p$ norms, ordered norms, and symmetric norms. Furthermore, we establish that submodular norms can be applied to optimization problems such as online facility location, stochastic probing, and generalized load balancing. This allows us to develop a logarithmic-competitive algorithm for online facility location with symmetric norms, to prove a logarithmic adaptivity gap for stochastic probing with symmetric norms, and to give an alternative poly-logarithmic approximation algorithm for generalized load balancing with outer $\ell_1$ norm and inner symmetric norms.