Carrier-selective passivating contacts for effective electron and hole extraction are crucial to the attainment of high efficiency in crystalline silicon (Si) solar cells. In this comprehensive review, the principle of carrier extraction and recombination mechanisms in conventional industrial Si solar cells are discussed first. Passivating contacts based on (i) amorphous hydrogenated Si and (ii) polysilicon/silicon oxide are next reviewed, with emphasis on carrier selectivity mechanisms including contact layer band alignment with silicon, and localized carrier transport in ultrathin oxides. More recent developments in dopant-free passivating contacts deposited by lower-cost fabrication processes with lower thermal budget are then described. This third category of non-Si based electron- and hole-selective passivating contacts include transition metal oxides, alkali/alkali earth metal fluorides and organic conjugated polymers. The photovoltaic performance of asymmetric double heterojunction Si solar cells fabricated using these non-Si passivating contacts and their stability in damp heat conditions are discussed and compared with Si based passivating contacts.