Floquet engineering is the concept of tailoring a system by a periodic drive, and it is increasingly employed in many areas of physics. Ultracold atoms in optical lattices offer a particularly large toolbox to design a variety of driving schemes. A strong motivation for developing these methods is the prospect to study the interplay between topology and interactions in a system where both ingredients are fully tunable. We review the recent successes of Floquet engineering in realizing new classes of Hamiltonians in quantum gases, such as Hamiltonians including artificial gauge fields, topological band structures and density-dependent tunnelling. The creation of periodically driven systems also gives rise to phenomena without static counterparts such as anomalous Floquet topological insulators. We discuss the challenges facing the field, particularly the control of heating mechanisms, which currently limit the preparation of many-body phases, as well as the potential future developments as these obstacles are overcome.