Numerical simulations of the collisional disruption of large asteroids show that although the parent body is totally shattered, subsequent gravitational reaccumulation leads to the formation of an entire family of large and small objects with dynamical properties similar to those of the parent body. Simulations were performed in two different collisional regimes representative of asteroid families such as Eunomia and Koronis. Our results indicate that all large family members must be made of gravitationally reaccumulated fragments; that the post-collision member size distribution and the orbital dispersion are steeper and smaller, respectively, than for the evolved families observed today; and that satellites form frequently around family members.