  Which reaction networks, when taken with mass-action kinetics, have the
capacity for multiple steady states? There is no complete answer to this
question, but over the last 40 years various criteria have been developed that
can answer this question in certain cases. This work surveys these
developments, with an emphasis on recent results that connect the capacity for
multistationarity of one network to that of another. In this latter setting, we
consider a network $N$ that is embedded in a larger network $G$, which means
that $N$ is obtained from $G$ by removing some subsets of chemical species and
reactions. This embedding relation is a significant generalization of the
subnetwork relation. For arbitrary networks, it is not true that if $N$ is
embedded in $G$, then the steady states of $N$ lift to $G$. Nonetheless, this
does hold for certain classes of networks; one such class is that of fully open
networks. This motivates the search for embedding-minimal multistationary
networks: those networks which admit multiple steady states but no proper,
embedded networks admit multiple steady states. We present results about such
minimal networks, including several new constructions of infinite families of
these networks.
