We consider two covering variants of the network design problem. We are given a set of origin/destination (O/D) pairs and each such O/D pair is covered if there exists a path in the network from the origin to the destination whose length is not larger than a given threshold. In the first problem, called the maximal covering network design problem, one must determine a network that maximizes the total demand of the covered O/D pairs subject to a budget constraint on the design costs of the network. In the second problem, called the partial covering network design problem, the design cost is minimized while a lower bound is set on the total demand covered. After presenting formulations, we develop a Benders decomposition approach to solve the problems. Further, we consider two different stabilization methods to determine the Benders cuts as well as the addition of cut-set inequalities to the master problem. Computational experiments show the efficiency of these different aspects.