Let $M^3$ be a differential manifold and $\gamma$ a non-negative integer. Then for or every metric $g$ on $M$, is it possible to find an area bound $C=C(g,\gamma)$ for closed embedded minimal surfaces of genus $\gamma$ in $(M,g)$. The answer is no in a rather strong sense. I will discuss the history and some solutions to this problem, in particular the difficulty which arises when $\gamma=0$.