A surface \(S\) is a closed 2-manifold, either orientable \((S_p\), of genus \(p\geq 0)\) or nonorientable \((N_p\), of genus \(p>0\)). A clean triangulation of \(S\) is a 2-cell imbedding of a graph \(G\) into \(S\) having every 2-cell bounded by a 3-cycle of \(G\) and having every 3-cycle of \(G\) bound a 2-cell. Then \(\tau(S)\) denotes the smallest possible number of triangles in a clean triangulation of \(S\). For example, the tetrahedron shows that \(\tau(S_0)=4\). (The imbedding of \(K_3\) into \(S_0\) is not considered, as it does not correspond to a polyhedron.) The new results are: \(\tau(S_1)=24\), \(\tau(N_1)=20\), and (for \(p\to\infty)\) \(\lim \tau(S_p)/p=4\), \(\lim \tau(N_p)/p=2\). Thus \(\tau(S)\) has the same asymptotic behavior as \(\delta(S)\) (the smallest number of triangles in a (not necessarily clean) triangulation of \(S\)); see \textit{M. Jungerman} and the second author [Acta Math. 145, 121--154 (1980; Zbl 0451.57005)] and the second author [Math. Ann. 130, 317--326 (1955; Zbl 0066.41702)].