A matroid on finite ground set E is a combinatorial abstraction of a function from E to a finite dimensional vector space. The first portion of this talk will cover basic constructs and results in the theory of matroids, and highlight some old (but powerful) applications in topology, optimization, and combinatorics. Of particular interest are the matroids represented by boundary operators of CW complexes; as we will see, invariants of these combinatorial objects encode useful information about the underlying spaces, and the specialization of several matroid algorithms to this context offers efficient partial solutions to problems, such as deciding embeddability, for which no viable alternatives are known (to us). We will close with a brief discussion on combinatorial k-connectivity, topological clustering, and proposed applications in robotics & neuroscience.