We investigate the timescales for stochasticity and chaotic mixing in a family of triaxial potentials that mimic the distribution of light in elliptical galaxies. Some of the models include central point masses designed to represent nuclear black holes. Most of the boxlike orbits are found to be stochastic, with mean Liapunov times that are 3 − 6 times the period of the long-axis orbit. In models with large cores or small black holes, the stochastic orbits mimic regular box orbits for hundreds of oscillations at least. However a small core radius or significant black hole mass causes most of the stochastic orbits to diffuse through phase space on the same timescale, visiting a significant fraction of the volume beneath the equipotential surface. Some stochastic orbits, with initial conditions lying close to those of regular orbits, remain trapped in all models. We estimate timescales for chaotic mixing in the more strongly stochastic models by evolving ensembles of 10^4 points until their distribution reaches a nearly steady state. Mixing initially takes place rapidly, with characteristic times of 10 − 30 dynamical times, as the phase points fill a region similar in shape to that of a box orbit. Subsequent mixing is slower, with characteristic times of hundreds of orbital periods. Mixing rates were found to be enhanced by the addition of modest force perturbations, and we propose that the stochastic parts of phase space might be efficiently mixed during the early phases of galaxy formation when such perturbations are large. The consequences for the structure and evolution of elliptical galaxies are discussed.