When taken with mass-action kinetics, which reaction networks admit multiple steady states? What structure must such a network possess? Mathematically, this question is: among certain parametrized families of polynomial systems, which families admit multiple positive roots (for some parameter values)? No complete answer is known, although various criteria now exist---some to answer the question in the affirmative and some in the negative. In this talk, we answer these questions for the smallest networks—those with only a few chemical species or reactions. Our results highlight the role played by the Newton polytope of a network (the convex hull of the reactant vectors). It has become apparent in recent years that analyzing this Newton polytope elucidates some aspects of the long-term dynamics and can be used to determine whether the network always admits at least one steady state. What is new here is our use of the geometric objects to determine whether a network admits steady state. Finally, our work is motivated by recent results that connect the capacity for multistationarity of a given network to that of certain related networks which are typically smaller: we are therefore interested in classifying small multistationary networks, and our results form the first step in this direction.