We calculate the Heegaard genus, h(M), of the closed non-orientable Seifert manifolds. If a 3-manifold M admits a decomposition M=H 1 ∪H 2 ∪H 3 into three orientable handlebodies of genera g 1 ,g 2 ,g 3 , respectively, and g 1 ≤g 2 ≤g 3 , we call the triple (g 1 ,g 2 ,g 3 ) the tri-genus of M if (g 1 ,g 2 ,g 3 ) is minimal among all such triples ordered lexicographically. We compute the tri-genus (g 1 ,g 2 ,g 3 ) of all non-orientable Seifert manifolds M which admit an S 1 -bundle structure with fiber an orientable surface. In this case the number g 3 is much bigger than h(M) for a fixed M. We obtain also that h(M) is an upper bound for the number g 3 in case M is a non-orientable Seifert manifold which does not admit an S 1 -bundle structure. We see that, although one could expect a relation between the number g 3 and the Heegaard genus, this relation, if any, can not be simple.