The use of kernel regression estimators is well known in the estimation of regression surfaces. The estimators involve a kernel with bandwidth h(≷0). The choice of h is important since a small h gives an estimator with a large variance, but if a large h is used then the bias is large. he bias is under specific smoothness assumptions, a functional of higher derivatives of the regression curve. Form a nonparametric viewpoint it is therefore desirable to choose the bandwidth in such a way that the variance and the bias are balanced independently of the smoothness of the curve. In this paper it is shown how such an asymptotically optimal h can be found. The construction of such an optimal bandwidth independent of the smoothness of the regression curve gives a positive answer to Question 3 of STONE'S (1982) paper. The proof only requires mild assumptions on the underlying density and the moments of the dependent variable y. An interesting relationship is discovered between the moments ofy and the smoothness of the kernel. The results of the present work extend that of STONE (1984) on kernel density estimation.