The Standard Model (SM) of particle physics has withstood thus far every attempt by experimentalists to show that it does not describe data. We discuss the SM in some detail, focusing on the mechanism of fermion mixing, which represents one of its most intriguing aspects. We discuss how this mechanism can be tested in b-quark decays, and how b decays can be used to extract information on physics beyond the SM. We review experimental techniques in b physics, focusing on recent results and highlighting future prospects. Particular attention is devoted to recent results from b decays into a hadron, a lepton and an anti-lepton, that show discrepancies with the SM predictions — the so-called B-physics anomalies — whose statistical significance has been increasing steadily. We discuss these experiments in a detailed manner, and also provide theoretical interpretation of these results in terms of physics beyond the SM.