The artificial compressibility method is designed to simulate two- and three-dimensional motions of viscous incompressible fluids. Three two-dimensional domains are studied in this paper: a square cavity with a uniform moving top, a rectilinear channel with a backward-facing step (flow over a sudden expansion) and double bent channels as well as one three-dimensional domain of a cubic cavity with a uniform moving top. The initial-boundary value problems for the systems of partial differential equations are reduced to the initial value problems for systems of ordinary differential equations, which are integrated using the backward-differentiation predictor-corrector method. Laminar flows have been performed for Re <= 2000 in the square and cubic cavities, for Re <= 800 over the backward-facing geometry and for Re <= 300 in the double bent channels.