This paper addresses the path planning problem for robotic applications using homotopy classes. These classes provide a topological description of how paths avoid obstacles, which is an added value to the path planning problem. Homotopy classes are generated and sorted according to a lower bound heuristic estimator using a method we developed. Then, the classes are used to constrain and guide path planning algorithms. Three different path planners are presented and compared: a graph-search algorithm called Homotopic A∗ (HA∗), a probabilistic sample-based algorithm called Homotopic RRT (HRRT), and a bug-based algorithm called Homotopic Bug (HBug). Our method has been tested in simulation and in an underwater bathymetric map to compute the trajectory of an Autonomous Underwater Vehicle (AUV). A comparison with well-known path planning algorithms has also been included. Results show that our homotopic path planners improve the quality of the solutions of their respective non-homotopic versions with similar computation time while keeping the topological constraints ; This research was sponsored by the Spanish Government under the grant DPI2011-27977-C03-02 and the TRIDENT EU FP7-Project under the Grant Agreement No. ICT-248497