A new monotonic and positive–definite filter is incorporated into an existing Semi-Lagrangian Inherently Conserving and Efficient (SLICE) scheme for transport problems in both Cartesian and spherical geometry. The SLICE scheme is based on a control-volume approach that uses multiple sweeps of a one-dimensional O (Δx4) conservation remapping algorithm along predetermined cascade directions. The new filter combines a selective detection algorithm, to pinpoint regions of non-monotonic behaviour, with a hierarchical reduction of the degree of the piecewise reconstruction in such regions, to re-establish monotonicity. The enhanced, monotonic and positive–definite, SLICE scheme is tested in one dimension, and then applied to standard two-dimensional test problems in both Cartesian and spherical geometries. Comparisons with published results of other conservative semi-Lagrangian schemes show that it performs well. © Crown copyright, 2005.