Reaction networks taken with mass-action kinetics arise in many settings, from epidemiology to population biology to systems of chemical reactions. The systematic study of the resulting polynomial ordinary differential equations began in the 1970s, and in recent years, this area has seen renewed interest, due in part to applications to systems biology. This talk focuses on the dynamics and steady states of such systems. Our main interest is in certain signaling networks, namely, multisite phosphorylation networks. In particular, these systems exhibit “toric steady states” (that is, the ODEs generate a binomial ideal), which enables us to efficiently determine their capacity for multiple steady states. Also, we show that when the phosphorylation/dephosphorylation mechanism is “processive” (binding of a substrate and an enzyme molecule results in addition or removal of phosphate groups at all phosphorylation sites), these systems exhibit rigid dynamics: each invariant set contains a unique steady state, which is a global attractor.