  Building on earlier work in the high energy and condensed matter communities,
we present a web of dualities in $2+1$ dimensions that generalize the known
particle/vortex duality. Some of the dualities relate theories of fermions to
theories of bosons. Others relate different theories of fermions. For example,
the long distance behavior of the $2+1$-dimensional analog of QED with a single
Dirac fermion (a theory known as $U(1)_{1/2}$) is identified with the $O(2)$
Wilson-Fisher fixed point. The gauged version of that fixed point with a
Chern-Simons coupling at level one is identified as a free Dirac fermion. The
latter theory also has a dual version as a fermion interacting with some gauge
fields. Assuming some of these dualities, other dualities can be derived. Our
analysis resolves a number of confusing issues in the literature including how
time reversal is realized in these theories. It also has many applications in
condensed matter physics like the theory of topological insulators (and their
gapped boundary states) and the problem of electrons in the lowest Landau level
at half filling. (Our techniques also clarify some points in the fractional
Hall effect and its description using flux attachment.) In addition to
presenting several consistency checks, we also present plausible (but not
rigorous) derivations of the dualities and relate them to $3+1$-dimensional
$S$-duality.
