This Letter describes the observation of the light-by-light scattering process, γγ→γγ, in Pb+Pb collisions at sNN=5.02 TeV. The analysis is conducted using a data sample corresponding to an integrated luminosity of 1.73 nb-1, collected in November 2018 by the ATLAS experiment at the LHC. Light-by-light scattering candidates are selected in events with two photons produced exclusively, each with transverse energy ETγ>3 GeV and pseudorapidity |ηγ|<2.4, diphoton invariant mass above 6 GeV, and small diphoton transverse momentum and acoplanarity. After applying all selection criteria, 59 candidate events are observed for a background expectation of 12±3 events. The observed excess of events over the expected background has a significance of 8.2 standard deviations. The measured fiducial cross section is 78±13(stat)±7(syst)±3(lumi) nb.