This paper studies the problem of how to space observations in regression so as to minimize the variance of an estimate of the regression function value at an arbitrary point in the domain of observations. Necessary and sufficient conditions are obtained for such a design, called a minimax design, in two dimensional polynomial regression of the type in which the regression function possesses a product structure. Such conditions are also obtained for minimax designs in one dimensional trigonometric and two dimensional spherical harmonics regression. Particular designs of the latter type are constructed.