The perimeter functional is known to oppose serious difficulties when it has to be handled within a topology optimization procedure. In this paper, a regularized perimeter functional Per(epsilon), defined for two- and three-dimensional domains, is introduced. On one hand, the convergence of Per(epsilon) to the exact perimeter when e tends to zero is proved. On the other hand, the topological differentiability of Per(epsilon) for epsilon > 0 is analyzed. These features lead to the design of a topology optimization algorithm suitable for perimeter-dependent objective functionals. Several numerical results illustrate the method.