Hydrogels, soft 3D materials of cross-linked hydrophilic polymer chains with a high water content, have found numerous applications in biomedicine because of their similarity to native tissue, biocompatibility and tuneable properties. In general, hydrogels are poor conductors of electric current, due to the insulating nature of commonly-used hydrophilic polymer chains. A number of biomedical applications require or benefit from an increased electrical conductivity. These include hydrogels used as scaffolds for tissue engineering of electroactive cells, as strain-sensitive sensors and as platforms for controlled drug delivery. The incorporation of conductive nanomaterials in hydrogels results in nanocomposite materials which combine electrical conductivity with the soft nature, flexibility and high water content of hydrogels. Here, we review the state of the art of such materials, describing the theories of current conduction in nanocomposite hydrogels, outlining their limitations and highlighting methods for improving their electrical conductivity.