Efforts to impart elasticity and multifunctionality in nanocomposites focus mainly on integrating polymeric1, 2 and nanoscale3, 4, 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 notches6, 7, 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 conductors3, 4, 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.