The past two decades has witnessed a dramatic burst of applications of topological thinking and theorems in the applied sciences, ranging from statistics to sensor networks, neuroscience, and more, to be surveyed here. Several challenges remain, including: (1) how to compute topological quantities efficiently; (2) how to extend the set of current applications and methods; and, perhaps most importantly, (3) how to educate end-users in the meaning and proper use of homological tools.This talk will demonstrate why homology is one of the most exciting new tools in applied mathematics.