In this talk I will review recent results on the development of a form factor program for integrable quantum field theories (IQFTs) perturbed by irrelevant operators. It has been known for a long time that under such perturbations integrability is preserved and that the two-body scattering phase gets deformed in a simple manner. The consequences of such a deformation are stark, leading to theories that exhibit a so-called Hagedorn transition and no UV completion. These phenomena manifest physically in several distinct ways. In our work we have mainly asked the question of how the deformation of the S-matrix translates into the correlation functions of the deformed theory. Does the scaling of correlators at long and short distances capture any of the "pathologies" mentioned above? Can our understanding of irrelevant perturbations tell us something about the space of IQFTs and about their form factors? In this talk I will answer these questions in the afirmative, summarising work in collaboration with Stefano Negro, Fabio Sailis and István M. Szécsényi.