In this article we consider the Keller–Segel model for chemotaxis on networks, both in the doubly parabolic case and in the parabolic–elliptic one. Introducing appropriate transition conditions at vertices, we prove the existence of a time global and spatially continuous solution for each of the two systems. The main tool we use in the proof of the existence result are optimal decay estimates for the fundamental solution of the heat equation on a weighted network.