Traditionally, scientists use one of a handful of function families (polynomials, Gaussians, exponentials, et al) to describe the underlying signal of an experiment corrupted with noise. However, with the ever-increasing influx of more data, flexible, but no less rigorous, shape descriptors are in high demand. Topological data analysis (TDA) uses classical constructions such as homology to provide a new science of shape. In this talk, I will survey some of the techniques and results of TDA with an eye towards understanding the following important question: How are the homologies of the fibers of a map f:X->S organized, and how can this be communicated to a non- mathematician?