In this thesis, we study gapless Quantum Field Theories using S-matrix bootstrap methods. These methods are intrinsically nonperturbative and utilize basic physical principles---such as Lorentz invariance, unitarity, and crossing symmetry---to constrain the space of allowed theories. After an introduction to the S-matrix and its properties, we present the basic ideas behind modern S-matrix bootstrap methods. The core of this manuscript is then divided into three parts.

In the first part, we use numerical methods to study the space of nonperturbative scattering amplitudes in the context of massless particles. We consider two cases: the scattering of a derivatively coupled massless scalar, and a massless spin-one particle---the photon. In both cases, we obtain constraints on low-energy observables and reconstruct explicit analytic, unitary, and crossing-symmetric amplitudes that saturate these bounds. These amplitudes are then analyzed, and we observe the emergence of various structures, such as Regge trajectories.

In the second part, we study nonperturbative properties of the gravitational scattering amplitude. Using the assumption that scattering at large distances is governed by semi-classical physics, we bound the Regge growth of the amplitude both at fixed t and smeared over it. Our key finding is that gravitational amplitudes admit dispersion relations with two subtractions. Then, using dispersion relations, we derive a bound on the local growth of the amplitude and a sum rule for the graviton pole.

In the last part, we leave the realm of nonperturbative physics and explore the space of weakly coupled stringy amplitudes. In addition to the usual constraints, we impose constraints from the shape of the leading Regge trajectory, which we assume to be linear. Using both a dual and a primal approach, we derive constraints on low-energy observables and observe where stringy amplitudes live in the space of all perturbative amplitudes. We exhibit a deformation of the Veneziano amplitude that satisfies all constraints at four points. We then attempt to extend this construction to higher points and discuss the challenges that arise.