An analytic theory of Coulomb scattering in particle beams is derived. It rests on the assumption that the N-body interaction can be well approximated by the vector superposition of all individual two-body interactions on a single, randomly chosen particle. This is the only assumption or approximation needed. A direct mathematical correspondence exists with the general problem of random flights, which was solved by Markov. The theoretical prediction agrees with Monte Carlo simulation to within a few percent for the lateral trajectory displacement for a particular case. The computation time required for the analytical result is at least an order of magnitude smaller than that required for Monte Carlo simulation.