In this talk, I will present recent advances on the Calderon problem for nonlocal wave equations, with particular emphasis on models incorporating a weak viscosity term. I will begin by reviewing inverse problems for viscous and damped nonlocal wave equations, and discuss the analytical tools used to establish unique determination results in each setting. I will then highlight the new challenges posed by weakly viscous equations and explain how these difficulties can be overcome.

A central part of the talk is devoted to a space–time Runge approximation theorem for such equations, which relies on the existence of very weak solutions to linear nonlocal wave equations with irregular sources. This approximation result plays a crucial role in the analysis of inverse problems for nonlinear weakly viscous wave models. At the end, I will present several illustrative applications. The results presented in this talk are based on joint work with Yi-Hsuan Lin, Teemu Tyni, and Katya Krupchyk.