CF-HiZELS, an ∼10 deg2 emission-line survey with spectroscopic follow-up: Hα, [O III] + Hβ and [O II] luminosity functions at z = 0.8, 1.4 and 2.2 We present results from the largest contiguous narrow-band survey in the near-infrared. We have used the wide-field infrared camera/Canada–France–Hawaii Telescope and the lowOH2 filter (1.187 ± 0.005 μm) to survey ≈10 deg2 of contiguous extragalactic sky in the SA22 field. A total of ∼6000 candidate emission-line galaxies are found. We use deep ugrizJK data to obtain robust photometric redshifts. We combine our data with the High-redshift(Z) Emission Line Survey (HiZELS), explore spectroscopic surveys (VVDS, VIPERS) and obtain our own spectroscopic follow-up with KMOS, FMOS and MOSFIRE to derive large samples of high-redshift emission-line selected galaxies: 3471 Hα emitters at z = 0.8, 1343 [O III] + Hβ emitters at z = 1.4 and 572 [O II] emitters at z = 2.2. We probe comoving volumes of >106 Mpc3 and find significant overdensities, including an 8.5σ (spectroscopically confirmed) overdensity of Hα emitters at z = 0.81. We derive Hα, [O III] + Hβ and [O II] luminosity functions at z = 0.8, 1.4, 2.2, respectively, and present implications for future surveys such as Euclid. Our uniquely large volumes/areas allow us to subdivide the samples in thousands of randomized combinations of areas and provide a robust empirical measurement of sample/cosmic variance. We show that surveys for star-forming/emission-line galaxies at a depth similar to ours can only overcome cosmic-variance (errors <10 per cent) if they are based on volumes >5 × 105 Mpc3; errors on L* and ϕ* due to sample (cosmic) variance on surveys probing ∼104 and ∼105 Mpc3 are typically very high: ∼300 and ∼40–60 per cent, respectively. 451 3 2303-2323 2303-2323 Oxford University Press