Topological Data Analysis (TDA) is a new approach to Data Analysis using techniques from Topology. One of the main methods in TDA known as Persistent (Co)Homology, keeps track of the evolution of topological features in a given dataset, such as clusters, loops, cavities and so on. Topological Data Analysis has recently been applied to many different research fields both within and outside mathematics (e.g. computer science and biology). In this talk I will introduce the Persistent Homology theory in TDA and then I will present (i) a joint work with M. Contessoto, F. Mémoli and L. Zhou on Persistent Cohomology Rings, (ii) a joint work with W. Kim and F. Mémoli on Dynamic Metric Spaces and also (iii) an ongoing joint work with P. Dlotko and K. Bartoszek on Phylogenetic Networks.