Detecting and understanding extragalactic Lyman [alpha] emission using 3D spectroscopy n this thesis we use integral field spectroscopy to detect and understand of Lyman α (Lyα) emission from high redshift galaxies. Intrinsically the Lyα emission at λ = 1216 Å is the strongest recombination line from galaxies. It arises from the 2p → 1s transition in hydrogen. In star forming galaxies the line is powered by ionisation of the interstellar gas by hot O and B stars. Galaxies with star formation rates of 1 10 Msol year are expected to have Lyα luminosities of 42 dex 43 dex (erg s) corresponding to fluxes ~ 17 dex 18 dex (erg s cm²) at redshifts z~3 where Lyα is easily accessible with ground based telescopes. However star forming galaxies do not show these expected Lyα fluxes. Primarily this is a consequence of the high absorption cross section of neutral hydrogen for Lyα photons σ ~ 14 dex (cm²). Therefore in typical interstellar environments Lyα photons have to undergo a complex radiative transfer. The exact conditions under which Lyα photons can escape a galaxy are poorly understood. ...