In practice, spatial data are sometimes collected at points (i.e. point-referenced data) and at other times are associated with areal units (i.e. block data). The change of support problem is concerned with inference about the values of a variable at points or blocks different from those at which it has been observed. In the context of block data which can be sensibly viewed as averaging over point data, we propose a unifying approach for prediction from points to points, points to blocks, blocks to points, and blocks to blocks. The approach includes fully Bayesian kriging. We also extend our approach to the the case of spatio-temporal data, wherein a judicious specification of spatio-temporal association enables manageable computation. Exemplification of the static spatial case is provided using a dataset of point-level ozone measurements in the Atlanta, Georgia metropolitan area. The dynamic spatial case is illustrated using a temporally extended version of this dataset, enabling comparison at the common time point.*To whom correspondence should be addressed.