Abstract: The first systematic study of deformation theory in algebraic number theory, specifically its application to the theory of Galois representations, was done by Barry Mazur (1987). The goal of this talk is to motivate why this is a useful and interesting thing to do. We begin with discussing why one should study Galois representations in the first place, let alone deform them. Then, we define appropriate categories that serve as the domains of our deformation functors and discuss aspects of their representability. Finally, we give examples of Galois representations arising “naturally” from arithmetic objects (like elliptic curves and modular forms) and from algebraic geometry (via the étale cohomology of smooth projective varieties). Time permitting, we will discuss some conjectures in the theory of Galois representations and the role deformation theory plays in understanding them better. No scheme theory is assumed.