In this talk, we will discuss Hilbert's syzygy theorem which tells us that any finitely generated graded module over a polynomial ring in n variables has a free resolution of length less than or equal to n. We will sketch a proof of this theorem and discuss some applications and related results. The talk will only assume a basic knowledge of commutative algebra and some homological algebra. Speaker(s): Reebhu Bhattacharyya (University of Michigan Ann Arbor)