We will discuss twistor theory in four-dimensional Minkowski space, and how it can be used to describe massless free fields of any integer or half-integer spin. We will then show how all-multiplicity scattering amplitudes for certain theories can be written down using twistor theory, with a particular focus on Yang-Mills theory and the Roiban-Spradlin-Volovich-Witten formula for its complete tree-level S-matrix.