The crossing number of a graph in a surface

The crossing number of $G$ in a surface $\sum$, denoted by $cr_{\sum}(G)$, is the minimum number of pairwise intersections of edges in a drawing of $G$ in the surface $\sum$. In particular, the crossing number of $G$ in the plane $S_0$ is denoted by $cr(G)$ for simplicity, which is an important notation to classify nonplanar graphs. In this talk, we will give some results concerning on the crossing number of certain graphs either in the plane or in the projective plane.