Unknotting number is a fundamental measure of how complicated a knot is, measuring how "far" it is from the unknot via crossing changes. It is a challenging invariant to compute, with a vast array of tools applied to its calculation, and many conjectures have grown up around it. In this talk we will discuss three such conjectures, each aimed at simplifying the task of computing unknotting numbers. We will describe how work on them led to a resolution of the "oldest" one: the additivity of unknotting number under connected sum.