In this work, the connected p-median and the connected facility location problems are defined. Applications arise in regional planning, design of telecommunications and transportation networks. For the first problem, two integer linear programming formulations are proposed. Adaptations are made in one of these formulations and are used to model the second problem. Approximation algorithms to solve the connected p-median problem are developed. A hybrid local search strategy is proposed. In order to speed up the local search iterations, ideas as circularity, first- improving strategy and discard neighbors are incorporated. A GRASP algorithm and a VNS heuristic are also proposed. A filter is used to reduce the computational time required and a path-relinking is applied to improve the results found. Computational experiments to compare the algorithms are reported. To improve these results, it is applied a post-optimization step to the GRASP and VNS heuristics.