ABSTRACT: Modular forms are central objects in number theory. They famously appear in the proof of Fermat's Last Theorem and more generally they play a role in the so-called Langlands program. This talk is concerned with the eigenforms, modular forms that are more algebraic in nature, and our ability to deform eigenforms into geometric families. My goal is to explain the history as well as the motivation for modular forms and families thereof. In addition, I will discuss recent results on the geometric properties of these families. This talk is intended for a general audience. A portion of this work is joint with David Hansen.