The gamma-ray spectroscopy of sixty excited states in ³⁹K below an excitation energy of 8.4 MeV has been studied. The data were obtained mainly from the ³⁶Ar+ alpha reaction at beam energies between 7.5 and 16.0 MeV, the remaining data coming from the ²⁴Mg+¹⁸O and ²⁸Si+¹⁶O reactions, both at beam energies of 42 MeV. 13 new levels were established in ³⁹K and several previously unobserved branches were revealed between the previously known levels. Spins, parities, mixing ratios and gamma-ray branching ratios were determined from angular distribution and linear polarisation measurements using the ³⁶Ar+ alpha reaction at several different beam energies. Mean lifetimes were deduced using both the Doppler-shift attenuation method (with the ³⁶Ar+ alpha reaction) and the recoil-distance method (with the ²⁸Si+¹⁶O reaction). The results have allowed transition strengths to be calculated for the decays between many of the levels. The results are discussed in terms of both the shell model and the simple weak coupling model.