High-spin states in Lu-165 were populated in the La-139(Si-30, 4n) reaction at a beam energy of 152 MeV and gamma-ray coincidences were measured with the EUROBALL spectrometer array. Nine new rotational bands were discovered, known band structures were considerably extended and many inter-band transitions were found. Structures with normal deformation coexist with bands associated with the strongly deformed triaxial energy minima found in calculations. Three of these triaxial bands form a family of wobbling excitations with phonon quanta n(w) = 0, 1 and 2. The wobbling mode is a unique signature of nuclear triaxiality. Configuration assignments are discussed for the observed band structures. An exchange of configuration between two of the new bands due to mixing is observed, resulting in different signature partnerships at low and high spins.