In this paper designs are found which are optimum for various models that include some autocorrelation in the covariance structure $V$. First it is noted that the ordinary least squares estimator is quite robust against small perturbations in $V$ from the uncorrelated case $V_0 = \sigma^2_I$. This "local" argument justifies our use of such estimators and restriction to the class of designs $\mathscr{X}^\ast$ (balanced incomplete block or Latin squares) optimum under $V_0$. Within $\mathscr{X}^\ast$ we search for designs for which the least squares estimator minimizes appropriate functionals of the dispersion matrix under various correlation models $V$. In particular, we consider "nearest neighbor" correlation models in detail. The solutions lead to interesting combinatorial conditions somewhat similar to those encountered in "repeated measurement" designs. Typically, however, the latter need not be BIBD's and require twice as many blocks. For Latin squares, and hypercubes, the conditions are less restrictive than those giving "completeness."