Rigorous Constraints, Bounds, and Relations for Scattering Amplitudes

We present a review of rigorous results on scattering amplitudes at low, intermediate, and high energies. The emphasis is on constraints that can be compared with experiment. Some new results are presented, in particular the existence of absolute bounds for inelastic processes, constraints for odd pion-pion waves, and a lower bound for scattering amplitudes at positive $t$. Most of the results rest only upon unitarity and what is popularly known as axiomatic analyticity, but there are a few cases where larger analyticity domains are needed.