They are used all the time in convergence proofs for a variety of different problems. If you can relate say a series or a sequence to a norm then you can use the results to show that something is bounded and thus converges.
This is particularly useful when you look at Hilbert spaces which are infinite-dimensional vector spaces. In finite dimensions convergence is obvious but in infinite dimensions, you get a lot of problems since many properties of convergence in finite spaces don't necessarily work in hilbert spaces.
Also note that you can apply norms to integrals since they are just an infinite summation as well.