This talk is based on a joint work with Magdalena Musat. We consider the following problem: When are the preduals of two hyperfinte factors (on separable Hilbert spaces) cb-isomorphic, i.e. isomorphic as operator spaces ? We show that if M is semifinite and N is Type III, then their preduals are not cb-isomorphic. Moreover we construct a one parameter family of 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. In contrast, Christensen and Sinclair proved in 1989 that all infinite dimensional hyperfinite factors with separable preduals are cb-isomorphic and more recently, Rosenthal, Sukochev and the first named author proved that all hyperfinite type IIIlambda factors, where 0 ¡ lambda =¡ 1, have cb-isomorphic preduals.