We have investigated the creation and destruction of asteroid families using a collisional code which follows the evolution of the size distributions of both the main-belt background population and individual families which are produced by collisions within the population. Starting with a small-mass initial belt, the number of families produced by collisional breakup of parent bodies larger than 100 km and surviving to the present time is in good agreement with the observed number of families. Increasing the mass of the initial belt increases the number of families in the model; hence the number of families that we see today provides a significant constraint on the overall collisional history of asteroids. Most families formed by disruption of large (D > 200 km) parent bodies are still recognizable today. Families formed from smaller parent bodies early in Solar System history have been eroded away, and there is a general trend in our models to have younger ages associated with families formed from smaller parent bodies. Today about one-third of the overall population should be in families, consistent with the findings of V. Zappal Γ et al. (1995, Icarus 116, 291-314).