I will give an introduction to current topics in the study of scattering amplitudes of gauge theory and gravity. There will be four two-hour lectures, and the plan is as follows. Lecture 1 will review basic modern techniques for scattering amplitudes, including recursion relations. Lecture 2 will present an overview of the relations describing gravity as a double copy of gauge theory, both for scattering amplitudes and for solutions to the equations of motion. Lecture 3 will introduce the formalism of the scattering equations, leading to the CHY formulas for amplitudes in theories of massless particles. Lecture 4 will present new developments in computing field-theory amplitudes from string-theory-inspired techniques, via a new type of worldsheet model known as ambitwistor string. This leads to an extension of the scattering equations to loop level.