Dynamical studies of elementary gas-phase bimolecular reactions have progressed significantly during the last few years owing to advancements in molecular beam and laser techniques as well as in theoretical methodologies. In this article we give a brief overview of the recent progress in the field of reaction dynamics and then survey recent work from our laboratory on reactions of atoms and radicals with simple molecules by the crossed molecular beam scattering method using mass-spectrometric detection. Emphasis is on three-atom (A + BC) and four-atom (AB + CD) reactions for which the interplay between experiment and theory is the strongest and the most detailed. Reactive differential cross-sections for the three-atom Cl + H2 and four-atom OH + H2 and OH + CO reactions are presented and compared with the results of quasiclassical and quantum-mechanical scattering calculations on ab initio potential-energy surfaces in an effort to assess the status of theory versus experiment. The reaction dynamics of electronically excited atoms are discussed too; the effect of electronic excitation on the reaction dynamics of atomic oxygen is examined using the reaction O(3P, 1D)+ H2S as an example.