Separation of variables and correlation functions: from spin chains to CFT

Recent years have seen great progress in developing and applying separation of variables (SoV) in quantum integrable models. I will describe the main results achieved in this program based on a series of papers with my collaborators. In particular, I will present the SoV construction for $\mathfrak{gl}(N)$ integrable spin chains. I will also show how to resolve the longstanding problem of computing the SoV measure, and how it leads to new highly compact determinant results for a large class of correlators and wavefunction overlaps. In addition, I will demonstrate the power of SoV in 4d integrable CFT's such as the fishnet theory. Lastly, I will outline highly promising applications in computation of exact correlators in $N=4$ super Yang-Mills theory.