Abstract algebra has a unifying and illuminating presence in many branches of computer science and electrical engineering, from logic and language theory to coding and criptography. In this talk, I will present an algebraic theory of dynamic network routing, with the goal of establishing fundamental results common to the various routing paradigms currently found in the Internet, ranging from intra-domain quality-of-service routing to inter-domain policy-based routing. The results concern the existence, uniqueness, and optimality of network potentials, together with the convergence of routing protocols to those network potentials. At heart, the algebraic constructs relevant to routing differ from semi-rings and dioides in that the distributive law is not required. This law is associated with optimality: its absence in our algebraic framework reflects Internet's inter-domain routing emphasis on global connectivity rather than on global optimality. Distance-vector and link-state protocols are the most common routing protocols in use in the Internet, the first in both inter- and intra-domain routing, the latter in intra-domain routing. I will show that distance-vector protocols and some of its variations operate correctly under very mild algebraic conditions whereas link-state protocols require the distributive law even for basic correctness. A great many applications arise as particular instances of the algebraic framework, and these will be shown throughout the talk.