In this article the Taylor-expansion method is introduced by which Monte Carlo ( (\mathrm{MC}) ) simulations in the canonical ensemble can be speeded up significantly. Substantial gains in computational speed of (20-40 %) over conventional implementations of the (M C) technique are obtained over a wide range of densities in homogeneous bulk phases. The basic philosophy behind the Taylor-expansion method is a division of the neighborhood of each atom (or molecule) into three different spatial zones. Interactions between atoms belonging to each zone are treated at different levels of computational sophistication. For example, only interactions between atoms belonging to the primary zone immediately surrounding an atom are treated explicitly before and after displacement. The change in the configurational energy contribution from secondary-zone interactions is obtained from the first-order term of a Taylor expansion of the configurational energy in terms of the displacement vector (d). Interactions with atoms in the tertiary zone adjacent to the secondary zone are neglected throughout. The Taylor-expansion method is not restricted to the canonical ensemble but may be employed to enhance computational efficiency of MC simulations in other ensembles as well. This is demonstrated for grand canonical ensemble (M C) simulations of an inhomogeneous fluid which can be performed essentially on a modern personal computer. 1995 Academic Press, Inc.