Underlying many biological models are chemical reaction networks (CRNs), and identifying allowed and forbidden dynamics in reaction networks may give insight into biological mechanisms. Algebraic approaches have been important in several recent developments. For example, computational algebra has helped us characterise all small mass action CRNs admitting certain bifurcations; allowed us to find interesting and surprising examples and counterexamples; and suggested a number of conjectures. Progress generally involves an interaction between analysis and computation: on the one hand, theorems which recast apparently difficult questions about dynamics as (relatively tractable) algebraic problems; and on the other, computations which provide insight into deeper theoretical questions. I’ll present some results, examples, and open
questions, focussing on two domains of CRN theory: the study of local bifurcations, and the study of multistationarity.