Abstract: When G is Green function on a domain, we say that the 3G theorem is true if G(x,z)G(z,w)/G(x,w) is less than c g(|x-z|) g(|z-w|) /g(|x-w|) where g(|x-y|) is Green function in R^d. 3G theorem above is closely related to Kato class, the conditional Gauge theorem and local Schrodinger operators. In this talk, we will discuss about the generalized 3G theorem; The generalized 3G form is G(x,y)G(z,w)/G(x,w) where y may be different from z. This 3G form appears in non-local Schrodinger operator theory. The generalized 3G theorem is equally important in non-local Schrodinger operators as is the classical 3G theorem in local Schrodinger operators. Using the generalized 3G theorem, we give a concrete form of non-local Kato class. We will also discuss other consequences of the generalized 3G theorem. In particular, some boundary potential theory of relativistic stable process in non-smooth open sets. This is a joint work with Young-Ran Lee.