We give an introduction to Topological Data Analysis (TDA), focusing on Persistent Homology. This is a technique to extract information about the shape of data and summarize it as a collection of intervals, also called a barcode. If the data is given in the form of a nice enough function on a smooth manifold, there are connections to Morse theory, where the topology of the underlying manifold is related to the critical points of the function. We then move the focus to Morse-Smale vector fields, which are a class of vector fields with good structural properties, and present new approaches to extend the aforementioned methods to these objects. We do so by applying different algebraic methods such as parametrized chain complexes and spectral sequences.