The enumeration of domino tilings of various shapes is a well-researched area in combinatorics. For the particular family of shapes called Aztec diamonds, the formula for the number of tilings is nice and simple - 2^(n(n+1)/2), for the Aztec diamond of size n. I will give a derivation of this formula using an algorithm that computes the sum of perfect matchings of the dual Aztec diamond graph, and prove that this algorithm indeed works. If time permits, I will discuss another algorithm that allows us to compute edge-inclusion probabilities, as well as how asymptotics of multivariate generating functions comes into play in this sub-area.