A closed point $x$ on a curve $C$ is called sporadic if $C$ has only finitely many closed points of degree at most $\deg(x)$. Bourdon, Ejder, Liu, Odumodu, and Viray have given a criterion for when images of sporadic points remain sporadic under an arbitrary morphism of curves. In this thesis we will study natural maps between the family of modular curves $X_0(n)$ and sporadic points of arbitrary degree. More specifically, we prove that non-cuspidal sporadic points on $X_0(n)$ corresponding to non-CM elliptic curves over number fields map to sporadic points on $X_0(d)$, for some bounded divisor $d$ of $n$.