We model the collisional evolution of Trojan asteroids using a numerical code which combines recent calculations of the intrinsic collision probabilities and impact speeds in the Trojan swarms (Marzari et al.1996) with our current understanding of the outcomes of high-velocity collisions between asteroid-sized bodies. Using plausible collision parameters and energy scaling of impact strength with size, we obtain a good match to the present Trojan population, as inferred by Shoemaker et al.(1989). The steep slope of the current Trojan size distribution at diameters larger than about 50-100 km is essentially unaltered by the collisional process and must reflect the formative processes of these bodies. At smaller sizes, collisions have produced a power law size distribution having a slope characteristic of collisionally relaxed populations. Hence, we cannot distinguish whether the small-size end of the distribution was formed before or after the disruptive collisional regime characteristic of the present Trojan environment was established. The formation of prominent dynamical families in the Trojan swarms is a natural outcome of the collisional process; their number may allow us to constrain the degree of collisional evolution that has occurred in the Trojans. Finally, we find that Trojan collisional debris escaping from the libration regions and ending up into cometary orbits could supply ≈10% of the current population of short-period comets and Centaur asteroids. Whether this occurs depends on the dynamical lifetime of such bodies and whether they contain enough volatiles to become active comets.