From a modern viewpoint the "S-matrix bootstrap" is the idea that general consistency conditions can be used to obtain quantitative constraints on scattering amplitudes. I will discuss the assumptions behind this approach, open questions about the structure of amplitudes, and discuss some fundamental results from the sixties and seventies. In the second part of the talk I will treat two modern approaches which were inspired by recent results on the conformal bootstrap, and show how they can be used to constrain scattering amplitudes in non-trivial ways.