Alexandrov spaces are a generalization of complete Riemannian manifolds with a lower sectional curvature bound. However, they may exhibit behaviors different from Riemannian manifolds due to the topological and metric singularities that they carry. It is then of fundamental importance to investigate whether one can extend a given property in the Riemannian setting to the Alexandrov setting. 
In this project we address some of these properties and explore how Alexandrov spaces behave with respect to each property. In the one direction, we focus our attention on the topological features. The primary objective here is to understand how far the topology of Alexandrov spaces are from that of the smooth manifolds. In the other direction, we examine Alexandrov spaces with positive curvature in the presence of symmetry and the goal is to classify them. To this end, we need to find obstructions and recognition tools, which, in particular, rely upon our understanding of topological behaviors of  Alexandrov spaces investigated in the former direction.