  We present numerical simulations of avalanches and critical phenomena
associated with hysteresis loops, modeled using the zero-temperature
random-field Ising model. We study the transition between smooth hysteresis
loops and loops with a sharp jump in the magnetization, as the disorder in our
model is decreased. In a large region near the critical point, we find scaling
and critical phenomena, which are well described by the results of an epsilon
expansion about six dimensions. We present the results of simulations in 3, 4,
and 5 dimensions, with systems with up to a billion spins (1000^3).
