Abstract: In this paper we give useful convergence criteria for Hermite-Fejer interpo-lation H_n(f, x) , One of them is: H_n(f,x) converges uniformly to f(x) if and only if the norms of H_n(f,x) are uniformly bounded for all n and H_n(x^k , x) converges uniformly to x^k for k=l,2.