Networks of chemical reactions have natural underlying combinatorial structure, allowing them to be represented as graphs or digraphs, perhaps with additional vertex or edge colourings/labellings. A variety of recent theorems demonstrate that finite computations on these graphs can determine the allowed dynamics of these systems to a surprising extent. For example, graph computations can lead to claims about the nature, uniqueness and stability of limit sets; the monotonicity of semiflows; the possibility of bifurcations; and so forth. Thus combinatorial approaches are making an increasing contribution to the study of allowed behaviours in these dynamical systems. I'll present a brief overview of some problems in this area, how these intersect with certain themes in analysis, convex geometry and combinatorics, but also how they suggest new mathematical questions of interest in their own right.