  The repulsive Hubbard model has been immensely useful in understanding
strongly correlated electron systems, and serves as the paradigmatic model of
the field. Despite its simplicity, it exhibits a strikingly rich phenomenology
which is reminiscent of that observed in quantum materials. Nevertheless, much
of its phase diagram remains controversial. Here, we review a subset of what is
known about the Hubbard model, based on exact results or controlled approximate
solutions in various limits, for which there is a suitable small parameter. Our
primary focus is on the ground state properties of the system on various
lattices in two spatial dimensions, although both lower and higher dimensions
are discussed as well. Finally, we highlight some of the important outstanding
open questions.
