Although it is straightforward to construct cubic splines in Cartesian geometry, this is not so for latitude-longitude grids over the sphere, because of the polar singularity. Previous work has either introduced ad hoc approximations over the polar caps, to the detriment of both continuity and accuracy, or has been restricted to interpolation of fields defined on uniform grids, with an even number of meridians, and with known polar values. These limitations are addressed herein by reformulating the construction of bicubic splines as the minimisation of an appropriate integral subject to certain constraints. © Crown Copyright 2010. Reproduced with the permission of HMSO. Published by John Wiley & Sons, Ltd.