In this paper we present a new routing paradigm that generalizes opportunistic routing for wireless multihop networks. In multirate anypath routing, each node uses both a set of next hops and a selected transmission rate to reach a destination. Using this rate, a packet is broadcast to the nodes in the set and one of them forwards the packet on to the destination. To date, there has been no theory capable of jointly optimizing both the set of next hops and the transmission rate used by each node. We solve this by introducing two polynomial-time routing algorithms and provide the proof of their optimality. The proposed algorithms run in roughly the same running time as regular shortest-path algorithms, and are therefore suitable for deployment in routing protocols. We conducted measurements in an 802.11b testbed network, and our trace-driven analysis shows that multirate anypath routing performs on average 80% and up to 6.4 times better than anypath routing with a fixed rate of 11 Mbps. If the rate is fixed at 1 Mbps instead, performance improves by up to one order of magnitude.