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A number field is monogenic (over $\mathbb{Q}$) if the ring of integers admits a power integral basis, i.e., a $\mathbb{Z}$-basis of the form $\{1, \alpha, \alpha^2, \ldots, \alpha^{n−1}\}$. The first portion of the talk will be spent revisiting some classical examples of monogeneity and non-monogeneity. We will pay particular attention to obstructions to monogeneity and relations to other arithmetic questions. With some of the classical context for monogeneity and power integral bases in hand, we will investigate the monogeneity of division fields of elliptic curves. This will culminate in two results: one describing non-monogeneity “horizontally” and the other non-monogeneity “vertically.” We will finish with some of the difficulties of generalizing to abelian varieties of dimension greater than one, highlighting a partial generalization.