We consider the problem of nonparametric regression when there are d explanatory variables which lie in a compact set  . Perhaps the most common example is two explanatory variables which lie in the unit square. We adapt kernel-type smoothers suitable for estimating a function of one variable to the estimation of a function of two or more variables. Kernel estimators are attractive because their explicit representation as a weighted local average makes them easy to implement and to understand. Especially in two dimensions, the ease and interpretability of these smoothers is an argument for their use. In addition, the theoretical tractability of kernel smoothers makes them attractive as components of more complicated estimation schemes; see for example Staniswalis (1989a), or Goldstein and Messer (1992). The radially symmetric kernel estimators we study here present a conceptually appealing approach to two issues which arise for kernel estimators in higher dimensions: first, what is the optimal shape of the window, and second, how to correct for boundary effects for very general domains