Abstract: Betti numbers refer to integer invariants of (suitably nice) topological spaces. Betti numbers also refer to integer invariants of (suitably nice) modules over (suitably nice) rings. In this talk, I will define each usage of the term “Betti number” and explain the relationship between them. Much of the talk will focus on two well-known conjectures: Halperin’s Toral Rank Conjecture and Avramov’s Total Betti Number Conjecture (which is derived from the Buchsbaum-Eisenbud-Horrocks Conjecture). In particular, I will describe some recent progress on these conjectures.