The isolated pentagon rule (IPR) is now widely accepted as a general rule for determining the stability of all-carbon fullerene cages composed of hexagons and pentagons. Fullerenes that violate this rule have been deemed too reactive to be synthesized. The stabilization of non-IPR endohedral fullerenes depends on charge transfer from the encapsulated metal clusters (endoclusters) to fullerene cages, the electronic properties of empty all-carbon cages, the matching size and geometries of fullerene and endocluster, as well as the strong coordination of the metal ions to fused pentagons. The stability of non-IPR exohedral fullerenes can be rationalized primarily by both the 'strain-relief' and 'local-aromaticity' principles. This Review focuses on recent work on stabilization of non-IPR fullerenes, including theoretical and empirical principles, experimental methods, and molecular structures of fused-pentagon fullerenes characterized so far. The special chemical properties of non-IPR fullerenes that distinguish them from IPR-satisfying ones are also emphasized.