Conformal field theory lies at the heart of two central topics in theoretical high energy physics: the study of quantum gravity and the mapping of quantum field theories through the renormalization group. In this thesis we explore a technique to study conformal field theories called Mellin amplitudes, which are essentially the Mellin transforms of conformal correlation functions.

The thesis is divided into two parts. In the first part we study the fundamental properties of Mellin amplitudes. We clarify the conditions for the existence of Mellin amplitudes nonperturbatively. We formulate a conjecture for the analytic properties of Mellin amplitudes and partially prove it. Finally, we discuss Polyakov conditions, which are nonperturbative zeros of Mellin amplitudes.

In the second part of the thesis we consider applications of Mellin amplitudes. We apply Mellin amplitudes to: the Wilson-Fisher model in 4-epsilon dimensions, three dimensional conformal field theories with slightly broken higher spin symmetry, two dimensional minimal models and loop diagrams in Anti-de-Sitter space. Our main result is the derivation of nonperturbative sum rules that constrain effective field theories in Anti-de-Sitter space.