Electrical conduction is well understood in materials formed from inorganic or organic building blocks, but their combination to produce conductive hybrid frameworks and networks is an emerging and rapidly developing field of research. Self-assembling organic-inorganic compounds offer immense potential for functionalising material properties for a wide scope of applications including solar cells, light emitters, gas sensors and bipolar transparent conductors. The flexibility of combining two distinct material classes into a single solid-state system provides an almost infinite number of chemical and structural possibilities; however, there is currently no systematic approach established for designing new compositions and configurations with targeted electronic or optical properties. We review the current status in the field, in particular, the range of hybrid systems reported to date and the important role of materials modelling in the field. From theoretical arguments, the Mott insulator-to-metal transition should be possible in semiconducting metal-organic frameworks, but has yet to be observed. The question remains as to whether electro-active hybrid materials will evolve from chemical curiosities towards practical applications in the near term.