A heteroclinic cycle in a dynamical system is given by a finite cyclic sequence of trajectories each connecting two fixed points. Robustness means that a cycle nearby still exists after perturbing the system in a particular admissible way. Here, one allows for a stable state with high symmetry to lose its stability and some of its symmetry. Or the perturbations applied let the system stay within a specified class of systems. Part I describes two typical representatives of the phenomenon. Part II surveys the results known so far on the theoretical side. Part III discusses applications to particular systems appearing as mathematical models in some context.