Many measurements and searches for physics beyond the standard model at the LHC rely on the efficient identification of heavy-flavour jets, i.e. jets originating from bottom or charm quarks. In this paper, the discriminating variables and the algorithms used for heavy-flavour jet identification during the first years of operation of the CMS experiment in proton-proton collisions at a centre-of-mass energy of 13 TeV, are presented. Heavy-flavour jet identification algorithms have been improved compared to those used previously at centre-of-mass energies of 7 and 8 TeV. For jets with transverse momenta in the range expected in simulated events, these new developments result in an efficiency of 68% for the correct identification of a b jet for a probability of 1% of misidentifying a light-flavour jet. The improvement in relative efficiency at this misidentification probability is about 15%, compared to previous CMS algorithms. In addition, for the first time algorithms have been developed to identify jets containing two b hadrons in Lorentz-boosted event topologies, as well as to tag c jets. The large data sample recorded in 2016 at a centre-of-mass energy of 13 TeV has also allowed the development of new methods to measure the efficiency and misidentification probability of heavy-flavour jet identification algorithms. The b jet identification efficiency is measured with a precision of a few per cent at moderate jet transverse momenta (between 30 and 300 GeV) and about 5% at the highest jet transverse momenta (between 500 and 1000 GeV).