Topological data analysis is an emerging concept in applied mathematics in which we characterize “shape of data” using topological methods. In particular, the persistent homology and its persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems including materials science, life science and social networks etc. In my talk, I will give a survey of these concepts both from mathematics and applications. If time is allowed, I will show several future challenges which deepen the mathematical understandings and possible applications.