Associated to every partially ordered set (poset) is a simplicial complex called the order complex. The order complex links combinatorics with topology and other areas of mathematics in a deep and fundamental way. Poset topology has applications in the theory of hyperplane and subspace arrangements, group theory, complexity theory, and knot theory. We will briefly describe some of these applications and give an overview of some of the main tools of the subject such as shellability and discrete Morse theory. As an illustration of these tools we establish connections between the homology of some interesting classes of posets.