It is well known that the maximum chromatic number of a graph on the o
rientable surface S-g is theta(g(1/2)). We prove that there are positi
ve constants c(1), c(2) such that every triangle-free graph on S-g has
chromatic number less than c(2)(g/log(g))(1/3) and that some triangle
-free graph on S-g has chromatic number at least c(1)g(1/3)/log(g). We
obtain similar results for graphs with restricted clique number or gi
rth on S-g or N-k. As an application, we prove that an S-g-polytope ha
s chromatic number at most O(g(3/7)). For specific surfaces we prove t
hat every graph on the double torus and of girth at least six is 3-col
orable and we characterize completely those triangle-free projective g
raphs that are not 3-colorable.