Following W.-Y. Hsiang's philosophy, in each category special attention should be given to the objects with \large amount of symmetry"". As demonstrated in the Riemannian setting, symmetry could provide a viable platform to develop new techniques and generate interesting examples on which to test conjectures or carry out geometric constructions. In this talk, I focus on cohomogeneity one actions on Alexandrov spaces, that is isometric actions whose orbit spaces are one dimensional, and present some results in this direction. I pay particular attention to the cohomogeneity one Alexandrov spaces with positive curvature and give a classi cation of such spaces in low dimensions. This is based on joint works with Fernando Galaz.