The stability of persistence diagrams is among the most important results in applied and computational topology but most results are with respect to the bottleneck distance between diagrams. This has two main implications: it makes the space of persistence diagrams rather pathological and it is often provides very pessimistic bounds with respect to outliers. In this talk I will discuss new stability results with respect to the p-Wasserstein distance between persistence diagrams. The main result is stability of persistence diagrams between different functions on the same finite simplicial complex in terms of the p-norm of the functions. This has applications to image analysis, persistence homology transforms and Vietoris-Rips complexes. This is joint work with Primoz Skraba.