We examine bounds on the locality of routing. A local routing algorithm makes a sequence of distributed forwarding decisions, each of which is made using only local information. Specifically, in addition to knowing the node for which a message is destined, an intermediate node might also know a) the subgraph corresponding to all network nodes within k hops of itself, for some value of k, b) the node from which the message originated, and c) which of its neighbours last forwarded the message. Our objective is to determine which of these parameters are necessary and/or sufficient to permit local routing as k varies on a network modelled by a connected undirected graph. In particular, we establish tight bounds on k for the feasibility of deterministic k-local routing for various combinations of these parameters, as well as corresponding bounds on dilation (the worst-case ratio of actual route length to shortest path length).