The theory governing the strong nuclear force-quantum chromodynamics-predicts that at sufficiently high energy densities, hadronic nuclear matter undergoes a deconfinement transition to a new phase of quarks and gluons(1). Although this has been observed in ultrarelativistic heavy-ion collisions(2,3), it is currently an open question whether quark matter exists inside neutron stars(4). By combining astrophysical observations and theoretical ab initio calculations in a model-independent way, we find that the inferred properties of matter in the cores of neutron stars with mass corresponding to 1.4 solar masses (M-circle dot) are compatible with nuclear model calculations. However, the matter in the interior of maximally massive stable neutron stars exhibits characteristics of the deconfined phase, which we interpret as evidence for the presence of quark-matter cores. For the heaviest reliably observed neutron stars(5,6) with mass M approximate to 2M(circle dot), the presence of quark matter is found to be linked to the behaviour of the speed of sound c(s) in strongly interacting matter. If the conformal bound cs2 <= 1/3 (ref. (7)) is not strongly violated, massive neutron stars are predicted to have sizable quark-matter cores. This finding has important implications for the phenomenology of neutron stars and affects the dynamics of neutron star mergers with at least one sufficiently massive participant.