Scattering amplitudes lie at the heart of quantum field theory, providing a powerful bridge between the deep structure of fundamental interactions and the practical demands of high-precision experiments. Remarkably, they exhibit hidden structures and an unexpected simplicity that goes far beyond what traditional Feynman-diagram methods would suggest. In this colloquium, I will present several examples of these structures and show how they can be exploited, particularly in the context of effective field theories. I will also describe applications ranging from high-energy particle collisions and basic questions in quantum gravity to precision predictions for gravitational-wave emission from binary black holes.