The Heat Balance Integral Method (HBIM) is generally applied to one-dimensional Cartesian heat flow and Stefan problems. The main reason for this being that solutions in spherical and cylindrical coordinates are less accurate than in Cartesian. Consequently, in this paper we examine the application of the HBIM to Stefan problems in spherical and cylindrical coordinates, with the aim of improving accuracy. The standard version as well as one designed to minimise errors will be applied on the original and transformed system. Results are compared against numerical and perturbation solutions. It is shown that for the spherical case it is possible to obtain highly accurate approximate solutions (more accurate than the first order perturbation for realistic values of the Stefan number). For the cylindrical problem the results are significantly less accurate.