For sparse tables unsmoothed cell proportions are known to be inappropriate estimators of the cell probabilities. Using information from neighboring cells, several smooth estimators of cell probabilities have been investigated (e.g. penalized likelihood, kernel and geometric combination estimators). Here we propose local polynomial smoothers to estimate the cell probabilities of a d-dimensional ordinal contingency table. Under minimal smoothness conditions, we obtain the convergence to zero of the mean sum of squared errors at the optimal rate. This result is valid since, in contrast to other existing nonparametric estimators, local polynomial smoothers have the same behaviour at boundary and interior cells. Smooth estimation of cell probabilities of a d-dimensional table requires the choice of a bandwidth matrix. So far only diagonal bandwidth matrices received attention. We consider a general d d matrix that permits smoothing in orientations different from the main directions. Some simulated examples illustrate that local polynomial smoothers with a general bandwidth matrix provide a very appealing alternative to already existing estimators.