Binary and multiple star systems may form by fragmentation, that is, through break-up of a dense molecular cloud core during the dynamical collapse phase that leads to the formation of protostellar objects. This review concentrates on theoretical models of fragmentation based on numerical hydrodynamical calculations in three spatial dimensions, using both finite-difference and smoothed particle hydrodynamics techniques. A variety of recent results are described, including calculations of the fragmentation of bar-like (prolate) clouds, fragmentation in clouds with initial power-law density and angular velocity distributions, tidally-induced fragmentation, fragmentation in cooling clouds, formation of hierarchical systems, and the dividing line between clouds that fragment and those that appear to form single protostars. A brief comparison of the predicted physical and dynamical properties of the theoretical fragments with the observed properties of main-sequence and pre-main-sequence binary stars lends supports to the hypothesis that fragmentation is the dominant formation mechanism for binary and multiple star systems. The major uncertainties regarding fragmentation are the extent to which precollapse clouds are susceptible to fragmentation, and the degree to which binary fragments undergo orbital decay and possibly mergers through interactions with the enveloping disk.