Although triple-lens microlensing has already been observed (OGLE-2006-BLG-109), a detailed theoretical study of three-body lensing is still missing. We investigate critical-curve topologies of three-point-mass gravitational microlenses, i.e., the number of critical-curve loops and their relative position. The general three-point-mass lens is described by five parameters: two mass ratios and three position parameters. Using a new numerical method for finding the conditions for topology changes of the critical curve, we map the full three-dimensional parameter space of a triple lens with fixed mass ratios in an arbitrary configuration. As an example, we present a full classification of lensing by three equal masses.