In deep penetration welding and cutting (by a laser or an electron beam) the surface of a metal is strongly evaporated and a hole develops. The maximum rate of penetration into a given material is simply shown to be proportional to beam power density. There is an initial pre-heating time before the surface starts to evaporate which for any given material depends upon power density (according to an inverse square law). The motion of the evaporating boundary is dominated and complicated by the discontinuity which latent heat represents. However, solutions can be found by using a perturbation expansion and these all tend to the evaporation-controlled limit. The maximum velocity is effectively achieved in about ten times the pre-heat time (depending upon the latent heat in relation to the normal heat capacity). For a stationary source (i.e. not traversing the surface) it is shown that energy considerations alone imply that the depth of the hole increases without limit.