Solar cells rely on the efficient generation of electrons and holes and the subsequent collection of these photoexcited charge carriers at spatially separated electrodes. High wafer quality is now commonplace for crystalline silicon (c-Si) based solar cells, meaning that the cell's efficiency potential is largely dictated by the effectiveness of its carrier-selective contacts. The majority of contacts currently employed in industrial production are based on highly doped-silicon, which can introduce negative side-effects including Auger recombination or parasitic absorption depending on whether the dopants are diffused into the absorber or whether they are incorporated into silicon layers deposited outside the absorber. Given the terawatt scale of deployment of c-Si solar cells, the search for alternative contacting schemes that can offer potential benefits in terms of performance, cost, ease of processing or stability is highly relevant. One such category of contacting schemes, with the potential to avoid the above mentioned issues, is that which employs metal compounds as the 'carrier-selective' layer. The last 7 years has seen a surge in interest on this topic and a few promising families of materials have emerged, most prominently the alkali/alkaline-earth metal compounds and the transition-metal oxides. The number of successful selective-contact demonstrations of materials within these families is fast increasing with the best solar cell demonstrations now exceeding 23%. However, in addition to improving their efficiency performance, several challenges remain if such contacts are to be considered for industrial adoption. These are mainly associated with poor stability, lack of compatibility with transparent electrodes and inability to be deposited using standard industrial techniques. This review covers the historical developments, current status and future prospects of metal-compound based selective contacts in the context of c-Si photovoltaics.