Light-by-light scattering (gamma gamma -> gamma gamma) is a quantum-mechanical process that is forbidden in the classical theory of electrodynamics. This reaction is accessible at the Large Hadron Collider thanks to the large electromagnetic field strengths generated by ultra-relativistic colliding lead ions. Using 480 mu b(-1) of lead-lead collision data recorded at a centre-of-mass energy per nucleon pair of 5.02 TeV by the ATLAS detector, here we report evidence for light-by-light scattering. A total of 13 candidate events were observed with an expected background of 2.6 +/- 0.7 events. After background subtraction and analysis corrections, the fiducial cross-section of the process Pb + Pb (gamma gamma) -> Pb-(center dot) + Pb-(center dot) gamma gamma, for photon transverse energy E-T > 3 GeV, photon absolute pseudorapidity vertical bar eta vertical bar < 2.4, diphoton invariant mass greater than 6 GeV, diphoton transverse momentum lower than 2 GeV and diphoton acoplanarity below 0.01, is measured to be 70 +/- 24 (stat.) +/- 17 (syst.) nb, which is in agreement with the standard model predictions.