We present new limits on an isotropic stochastic gravitational-wave background (GWB) using a six pulsar data set spanning 18 yr of observations from the 2015 European Pulsar Timing Array data release. Performing a Bayesian analysis, we fit simultaneously for the intrinsic noise parameters for each pulsar, along with common correlated signals including clock, and Solar system ephemeris errors, obtaining a robust 95 per cent upper limit on the dimensionless strain amplitude A of the background of A < 3.0 × 10⁻¹⁵ at a reference frequency of 1 yr⁻¹ and a spectral index of 13/3, corresponding to a background from inspiralling supermassive black hole binaries, constraining the GW energy density to Ω_gw(f)h² < 1.1 × 10⁻⁹ at 2.8 nHz. We also present limits on the correlated power spectrum at a series of discrete frequencies, and show that our sensitivity to a fiducial isotropic GWB is highest at a frequency of ∼5 × 10⁻⁹ Hz. Finally, we discuss the implications of our analysis for the astrophysics of supermassive black hole binaries, and present 95 per cent upper limits on the string tension, Gμ/c², characterizing a background produced by a cosmic string network for a set of possible scenarios, and for a stochastic relic GWB. For a Nambu–Goto field theory cosmic string network, we set a limit Gμ/c² < 1.3 × 10⁻⁷, identical to that set by the Planck Collaboration, when combining Planck and high-ℓ cosmic microwave background data from other experiments. For a stochastic relic background, we set a limit of , a factor of 9 improvement over the most stringent limits previously set by a pulsar timing array.