In particle physics, scattering amplitudes are the key ingredients for calculation of cross-sections measured at collider experiments. Over the last two decades, multiple research efforts have explored alternatives to the conventional Feynman diagrammatic approach to scattering amplitudes. This has resulted in the development of new efficient calculational tools and in novel insights. In the colloquium, I will review the modern approach to amplitudes. Then I'll turn the process on its head: instead of using amplitudes as a way to extract the physics from a given theoretical model, I will discuss how the mathematical properties of scattering amplitudes allow us to place fundamental bounds on the couplings of particle interactions in low-energy effective field theories.