For each complex number c, the divisor function \sigma_c is the arithmetic function given by \sigma_c(n)=\sum_{d|n}d^c. We will touch upon a small sliver of the rich and exhilarating history behind these classical functions, leading into a discussion of recent questions concerning their ranges. More specifically, we consider the basic topological properties of the closure of the range of \sigma_c. This talk will provide a summary of some of my own work along with some recent results due to Carlo Sanna and Nina Zubrilina. We will also present a few open problems and conjectures.