There have been significant recent advances in realizing bandstructures with geometrical and topological features in experiments on cold atomic gases. We provide an overview of these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. We describe the different methods that have been used to generate these novel bandstructures for cold atoms, as well as the physical observables that have allowed their characterization. We focus on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. However, we also describe how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, we describe the forms of inter-particle interactions that emerge when atoms are subjected to these energy bands, and some of the many-body phases that may be sought in future experiments.
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