The paper describes fluidic networks consisting of individually controlled branches. The networks' basic building blocks are conduits equipped with two electrodes positioned on opposing walls. The entire device is either subjected to an external uniform magnetic field or fabricated within a magnetic material. When a prescribed potential difference is applied across each electrode pair, it induces current in the liquid (assumed to be at least a weak electrolyte solution). Analogously with electric circuits, by judicious application of the potential differences at various branches, one can direct liquid flow in any desired way without a need for mechanical pumps or valves. Equipped with additional, internally located electrodes, the network branches double as stirrers capable of generating chaotic advection. The paper describes the basic building blocks for such a network, the operation of these branches as stirrers, a general linear graph-based theory for the analysis and optimal control of fluidic magneto-hydrodynamic networks, an example of a network fabricated with low temperature, co-fired ceramic tapes, and preliminary experimental observations that illustrate that the ideas described in this paper can, indeed, be implemented in practice.