Though rooted in pure mathematics, algebraic topology has recently enabled novel analysis techniques for biological data. Specifically, the method called persistent homology which employs ideas from topology to characterize the voids in a complex network. In this talk, I will briefly introduce persistent homology and give an overview of how we have recently applied this tool to answer questions in neuroscience and semantic learning. Finally, I will provide additional examples of topology in biology and suggest how current research in pure algebraic topology may soon aid in our understanding of complex systems.