In 1989 Christensen and Sinclair proved that all infinite hyperfinite (= injective) factors with separable preduals are cb-isomorphic (i.e., isomorphic as operator spaces). When looking at preduals, the situation turns out to be very different. In recent work with Uffe Haagerup, we show that if M and N are hyperfinite factors (on separable Hilbert spaces) such that M is semifinite and N is of type III, then their preduals are not cb-isomorphic. Furthermore, we construct a one-parameter family of hyperfinite type III0 factors with mutually non-cb-isomorphic preduals, and we give a characterization of those hyperfinite factors M whose preduals are cb-isomorphic to the predual of the hyperfinite type III1 factor.