Ultracold neutral plasmas occupy an exotic regime of plasma physics in which electrons form a swarming, neutralizing background for ions that sluggishly move in a correlated manner. Strong interactions between the charged particles give rise to surprising dynamics such as oscillations of the average kinetic energy during equilibration and extremely fast recombination. Such phenomena offer stimulating and challenging problems for computational scientists, and the physics can be applied to other environments, such as the interior of gas giant planets and plasmas created by short-pulse laser irradiation of solid, liquid, and cluster targets.