  The interaction between light and matter can give rise to novel topological
states. This principle was recently exemplified in Floquet topological
insulators, where \emph{classical} light was used to induce a topological
electronic band structure. Here, in contrast, we show that mixing \emph{single}
photons with excitons can result in new topological polaritonic states --- or
"topolaritons". Taken separately, the underlying photons and excitons are
topologically trivial. Combined appropriately, however, they give rise to
non-trivial polaritonic bands with chiral edge modes allowing for
unidirectional polariton propagation. The main ingredient in our construction
is an exciton-photon coupling with a phase that winds in momentum space. We
demonstrate how this winding emerges from spin-orbit coupling in the electronic
system and an applied Zeeman field. We discuss the requirements for obtaining a
sizable topological gap in the polariton spectrum, and propose practical ways
to realize topolaritons in semiconductor quantum wells and monolayer transition
metal dichalcogenides.
