Let $a,b,$ and $c$ be relatively prime positive integers such that $a+b=c$. How does c compare to $\operatorname{rad}(abc)$, where rad(n) denotes the product of the distinct prime factors of $n$? According to the explicit $abc$ conjecture, it is always the case that $c$ is less than the square of $\operatorname{rad}(abc)$. This simple statement is incredibly powerful, and as a consequence, one gets a (marginal) proof of Fermat's Last Theorem for exponent $n$ greater than $5$. In this talk, we introduce Masser and Oesterlé's $abc$ conjecture and discuss some of its consequences, as well as some of the numerical evidence for the conjecture. We will then introduce elliptic curves and see that the $abc$ conjecture has an equivalent formulation in this setting, namely, the modified Szpiro conjecture. We conclude the talk by discussing a recent result that establishes the existence of sharp lower bounds for the modified Szpiro ratio of an elliptic curve that depends only on its torsion structure.