In this paper a comprehensive theoretical analysis of the two-neutrino (2ν2β) and neutrinoless (0ν2β) doublebeta decays of 78Kr is performed by evaluating the corresponding nuclear matrix elements by combining the quasiparticle random-phase approximation with the multiple-commutator model. Transitions to the ground state 0+ gs and 0+ and 2+ excited states in 78Se are investigated by using G-matrix-based nuclear forces. The channels β+β+, β+EC, and ECEC are discussed for the 2ν2β decays and the channels β+β+ and β+EC for the 0ν2β decays. The associated half-lives are computed to see if the detection of some of these transitions is experimentally feasible.