In this talk, I would like to present a view of how random walks can serve as the bridge between quantitative features of data, which are determined by geometric tools such as curvature, and qualitative features that can be detected by topological methods.  After presenting some of the main ideas of geometric and topological data analysis via random walks on graphs, I will talk about the main challenges of extending these ideas to higher dimensions and I will discuss the known results as well as open problems in both theory and applications.