Minimal surfaces with genus zero

A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known classification results for properly embedded minimal surfaces with genus zero in $\mathbb{R}^3$ or quotients of $\mathbb{R}^3$ by one or two independent translations. This does not intend to be an exhaustive review of the tools or proofs in the field, but a simple explanation of the currently known results.