I will discuss several interesting applications of homology and cohomology in visualization. First, I will describe how the topological notion of robustness, a relative of persistence, could be applied in the analysis and visualization of vector fields. In particular, robustness allows us to quantify the stability of critical points, to investigate how the stability of a critical point evolves over time and to infer correspondences between features. It also leads to novel vector field simplification schemes in 2D and 3D. Second, I will discuss detecting circular and branching structures in high dimensions based on cohomology parametrization, and its application in software visualization. Joint work with Primoz Skraba, Paul Rosen, Guoning Chen, Harsh Bhatia, Valerio Pascucci, Brian Summa, A.N.M. Imroz Choudhury and Mikael Vejdemo-Johansson.