Top quark production in the forward region in proton-proton collisions is observed for the first time. The W+b final state with W→μν is reconstructed using muons with a transverse momentum, p_{T}, larger than 25 GeV in the pseudorapidity range 2.0<η<4.5. The b jets are required to have 50<p_{T}<100 GeV and 2.2<η<4.2, while the transverse component of the sum of the muon and b-jet momenta must satisfy p_{T}>20 GeV. The results are based on data corresponding to integrated luminosities of 1.0 and 2.0 fb^{-1} collected at center-of-mass energies of 7 and 8 TeV by LHCb. The inclusive top quark production cross sections in the fiducial region are σ(top)[7 TeV]=239±53(stat)±33(syst)±24(theory) fb,σ(top)[8 TeV]=289±43(stat)±40(syst)±29(theory) fb.These results, along with the observed differential yields and charge asymmetries, are in agreement with next-to-leading order standard model predictions.