We begin with a introduction to the notion of a resolution of a module over a Noetherian ring, leading to Betti numbers over local or graded rings, and some problems related to them. Most of the talk will focus on the graded case. One of the recent developments in this area is the resolution of the Boij-Soderberg conjectures by Eisenbud-Schreyer (2009). We discuss the motivation behind the conjectures, with a quick word on the techniques used in their resolution. If time permits, we will see other scenarios where parts of the Boij-Soderberg conjectures hold, and discuss obstacles in extending the Eisenbud-Schreyer techniques in general. This last part is joint work with Rajiv Kumar.