This article reviews recent progress in our understanding of gas-phase neutral reaction dynamics as made possible by improvements in the crossed molecular beam scattering technique for measuring reactive differential cross sections. A selection of crossed-beam studies on systems that play a fundamental role in our basic understanding of reaction phenomena are discussed to illustrate the capabilities of the experimental method. The examples include benchmark atom-diatom abstraction and insertion reactions, and four-atom radical reactions for which state-to-state, state-resolved, or state-averaged differential cross sections have recently been measured. The results are discussed in the light of the latest related theoretical developments regarding the treatment of potential energy surfaces and the dynamics of the systems. Recent results on crossed-beam studies of chemically relevant reactions of carbon, nitrogen, and oxygen atoms are also reviewed, and the latest developments in the technique are noted.