Charge migration along DNA molecules attracted scientific interest for over half a century. Reports on possible high rates of charge transfer between donor and acceptor through the DNA, obtained in the last decade from solution chemistry experiments on large numbers of molecules, triggered a series of direct electrical transport measurements through DNA single molecules, bundles and networks. These measurements are reviewed and presented here. From these experiments we conclude that electrical transport is feasible in short DNA molecules, in bundles and networks, but blocked in long single molecules that are attached to surfaces. The experimental background is complemented by an account of the theoretical/computational schemes that are applied to study the electronic and transport properties of DNA-based wires. Examples of selected applications are given, to show the capabilities and limits of current theoretical approaches to accurately describe the wires, interpret the transport measurements, and predict suitable strategies to enhance the conductivity of DNA nanostructures.