Stoichiometrically exact and potentially feasible catalytic or metabolic pathways can be found by synthesizing the networks of plausible elementary or metabolic reactions constituting Such pathways, respectively. The Current contribution presents a mathematically exact algorithmic approach for carrying Out the necessary synthesis, which is profoundly complex combinatorially. The approach is based on the unique. graph-representation in terms of P-graphs (process graphs), a set of axioms, and a Group of combinatorial algorithms. The inclusion or exclusion of a step of each elementary or metabolic reaction in the pathway of interest hinges oil the general combinatorial properties of feasible reaction networks. At the outset, a brief overview is given of successful applications to date, followed by an Outline of the methodology, on which the approach is based. The approach is illustrated by implementing it to three new examples comprising two catalytic reactions, catalytic combustion of hydrogen and reduction of nitrogen oxide, and one metabolic reaction, involved in the production of ethanol by yeast. The efficacy of the approach is discussed in light of the results obtained from these examples. Finally, a brief discourse is given of our current and future efforts.