The study of the Rayleigh-Taylor (RT) instability, in which a heavy fluid initially lies on top of a lighter fluid in the presence of a vertical gravitational field, has wide application to many diverse fields, such as astrophysics, oceanography, and inertial confinement fusion. Furthermore, it serves as an archetype for more general instances of turbulent mixing. The RT instability has been studied extensively by both theory and experiment, as well as by simulation. Until recently, most of simulation have been carried out using continuum (i.e. Navier-Stokes-based) methods. However, given the existence of many regimes in which continuum methods break down (e.g. small-scale flows, or flows with a high Knudsen number) it is useful to consider a more fundamental approach based on particle methods. Though generally more computationally intensive, such methods have become feasible in recent years as increases in large-scale computational capacity have allowed multi-billion-particle simulations to be performed. In this work, we use Direct Simulation Monte Carlo (DSMC), a fast, stochastic, atomistic algorithm, to simulate the RT instability. This allows us to capture numerous physical effects not resolved by standard continuum methods, such as the discontinuous breakup of flow features, and the effects of micro-scale fluctuations. In addition, we compare with continuum predictions such properties as the initial growth spectrum of the interface, as well as the development in time of the mixing zone width.