Efforts to impart elasticity and multifunctionality in nanocomposites focus mainly on integrating polymeric(1,2) and nanoscale(3-5) components. Yet owing to the stochastic emergence and distribution of strain-concentrating defects and to the stiffening of nanoscale components at high strains, such composites often possess unpredictable strain-property relationships. Here, by taking inspiration from kirigami-the Japanese art of paper cutting-we show that a network of notches(6-8) made in rigid nanocomposite and other composite sheets by top-down patterning techniques prevents unpredictable local failure and increases the ultimate strain of the sheets from 4 to 370%. We also show that the sheets' tensile behaviour can be accurately predicted through finite-element modelling. Moreover, in marked contrast to other stretchable conductors(3-5), the electrical conductance of the stretchable kirigami sheets is maintained over the entire strain regime, and we demonstrate their use to tune plasma-discharge phenomena. The unique properties of kirigami nanocomposites as plasma electrodes open up a wide range of novel technological solutions for stretchable electronics and optoelectronic devices, among other application possibilities.