Traditional approaches to network design separate the issues of designing the network itself and designing its management and control subsystems. This paper proposes an approach termed routing-oriented network design, which is based on designing the network topology and its routing scheme together, attempting to optimize some of the relevant parameters of both simultaneously. This approach is explored by considering the design of communication networks supporting efficient routing in the special case of points located in the Euclidean plane. The desirable network parameters considered include low degree and small number of communication links. The desirable routing parameters considered include small routing tables, small number of hops and low routing stretch. Two rather different schemes are presented, one based on direct navigation in the plane and the other based on efficient hierarchical tree covers. On a collection of n sites with diameter D, these methods yield networks with maximum degree O(log D) (hence a total of O(n log D) communication links), coupled with routing schemes with constant routing stretch, O(log n log D) memory bits per vertex and routes with at most log n or log D hops.