Top quark production in the forward region in proton-proton collisions is observed for the first time. The W + b final state with W -> mu nu is reconstructed using muons with a transverse momentum, p(T), larger than 25 GeV in the pseudorapidity range 2.0 < eta < 4.5. The b jets are required to have 50 < p(T) < 100 GeV and 2.2 < eta < 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 sigma(top)[7 TeV] = 239 +/- 53(stat) +/- 33(syst) +/- 24(theory) fb; sigma(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.