This talk will review a large class of explicit models in quantum field theory that can be formulated in terms of a "twist" (a certain operator on a two-particle Hilbert space satisfying a subtle positivity condition) and a localization structure formulated in terms of modular theory. In certain models, the twist takes the role of an elastic two-body S-matrix, whereas in others, it can be viewed as being induced by a noncommutative background, and in yet others, it connects to free probability. We thus see several of Harald Grosse's prime research interests meeting here, and indeed some of the models discussed originate in joint work with him. The interplay of the twist and the localization structure leads to a question of the vacuum vector being separating for a certain algebra of field operators and can be used to derive two fundamental concepts of exactly solvable models, namely the Yang-Baxter equation and crossing symmetry.