  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.
