In the first part of this paper, we review the developments which led to the present state-of-the-art in the surface passivation of today's industrially predominant dopant-diffused crystalline silicon (c-Si) solar cells, based on dielectric layers such as silicon oxide, silicon nitride, aluminum oxide and stacks thereof. In the second part of this review, we focus on the future developments in the field of c-Si solar cells based on carrier-selective passivation layers. Whereas the dielectric layers are insulating and are hence applied only for passivating the non contacted areas of the silicon surface, the carrier-selective passivation layers are intended to provide an effective passivation of non-contacted as well as contacted areas of a c-Si solar cell, thereby increasing the efficiency potential of c-Si solar cells significantly. Due to the fact that the carrier-selective layers are implemented in a contact, besides the good passivation properties for minorities, these layers must also provide a good majority-carrier transport, i.e. they have to provide a low contact resistance. Both properties, i.e. suppression of minority carrier recombination as well as good majority-carrier transport, define the selectivity of the carrier-selective contact, which is an important figure of merit for the assessment and comparison of different types of carrier selective contacts. One very promising type of carrier-selective passivation layer is based on heavily doped polycrystalline silicon layers deposited on a thin silicon oxide layer, the latter providing the excellent passivation while enabling efficient majority-carrier transport via pin-holes and/or tunneling. Moreover, we discuss metal oxides and conductive polymers, which have only recently been applied to c-Si photovoltaics, but seem to have a promising potential as low-cost selective contact materials. We finally compare combinations of the various options of carrier-selective layers concerning their combined selectivities and efficiency potentials.