  Quantum spin liquids form a novel class of matter where, despite the
existence of strong exchange interactions, spins do not order down to the
lowest measured temperature. Typically, these occur in lattices that act to
frustrate the appearance of magnetism. In two dimensions, the classic example
is the kagome lattice composed of corner sharing triangles. There are a variety
of minerals whose transition metal ions form such a lattice. Hence, a number of
them have been studied, and were then subsequently synthesized in order to
obtain more pristine samples. Of particular note was the report in 2005 by Dan
Nocera's group of the synthesis of herbertsmithite, composed of a lattice of
copper ions sitting on a kagome lattice, which indeed does not order down to
the lowest measured temperature despite the existence of a large exchange
interaction of 17 meV. Over the past decade, this material has been extensively
studied, yielding a number of intriguing surprises that have in turn motivated
a resurgence of interest in the theoretical study of the spin 1/2 Heisenberg
model on a kagome lattice. In this colloquium article, I will review these
developments, and then discuss potential future directions, both experimental
and theoretical, as well as the challenge of doping these materials with the
hope that this could lead to the discovery of novel topological and
superconducting phases.
