A curable composite material having high z-direction electrical conductivity. The curable composite material includes two or more layers of reinforcement carbon fibers that have been infused or impregnated with a curable matrix resin and an interlaminar region containing at least conductive nano-sized particles, e.g. carbon nanotubes, and a light-weight carbon veil. According to another embodiment, the interlaminar region further contains polymeric toughening particles. Methods for fabricating composite materials and structures are also disclosed.