Abstract: A near-triangular embedding is a graph embedded into some surface whose facial walks but one are 3-gons. In this paper the authors show that if a graph G is a triangulation of the sphere S₀ (or the torus S₁), then Ghas a near-triangular embedding into S_k for k = h,h+1,\cdots ,\lfloor\frac{\beta(G)}{2}\rfloor$, where h = 0(or 1) and $\beta(G)$ is the Betti number of G.

Key words: Triangulation, Embedding, Orientable surface