Stable maps of surfaces into the plane

In this paper we investigate Σ1,0-maps of closed surfaces into the plane, specifically, the singular sets of such maps. This set is the disjoint union of finitely many embedded circles in the surface; we will determine all possible numbers of components for each surface. During this survey we will construct singular maps of all closed surfaces into the plane which are simplest in the sense that they have the least possible number of cusps (0 or 1) and under this condition their singular sets have the least possible number of components (1 or 2). Additionally, we will provide a simplified and shortened proof of the dimension 2 case of the theorem concerning the elimination of cusps (due to Millett, and Levine for the higher-dimensional cases).