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Many wireless mesh networks are based on unicast routing protocols even though those protocols do not provide a particularly good fit for such scenarios. In this article, we report about an alternative routing paradigm, tailor-made for large multihop wireless mesh networks: field-based anycast routing. In particular, we present HEAT, a routing protocol based on this paradigm. In contrast to previous protocols, HEAT requires communication only between neighboring nodes. The underlying routing concept is a field similar to a temperature field in thermal physics. In extensive simulation experiments, we found that HEAT has excellent scalability properties due to a fully distributed implementation, and it provides much more robust routes than the unicast protocols, AODV and OLSR. As a consequence, in large-scale mobile scenarios, the packet delivery ratio with HEAT is more than two times higher, compared to AODV or OLSR. These promising results indicate that HEAT is suitable for large-scale wireless mesh networks that cover entire cities.