The flavour-tagging algorithms developed by the ATLAS Collaboration and used to analyse its dataset of $\sqrt{s} = 13$ TeV pp collisions from Run 2 of the Large Hadron Collider are presented. These new tagging algorithms are based on recurrent and deep neural networks, and their performance is evaluated in simulated collision events. These developments yield considerable improvements over previous jet-flavour identification strategies. At the 77% b-jet identification efficiency operating point, light-jet (charm-jet) rejection factors of 170 (5) are achieved in a sample of simulated Standard Model $t\bar{t}$ events; similarly, at a c-jet identification efficiency of 30%, a light-jet (b-jet) rejection factor of 70 (9) is obtained.