Four-wave and six-wave mixing processes under two-photon transitions (4S-8S), (4S-7S), (4S-6D) and (4S-5D) in potassium vapour have been investigated for the wavelength range 300-370nm. The wave-coupling schemes and the non-collinear phase-match angles for the observed UV lines have been analyzed. Frequency doubling under resonant and near-resonant conditions has been observed.