In this paper we study the real point spectrum of the multigroup neutron transport operator for a large class of unsymmetric collision matrices in bounded spatial domains. Necessary and sufficient conditions for existence, finiteness and sufficient conditions for infiniteness of the real point spectrum are given. Estimates of the number of real eigenvalues and localization results are also provided. The last part of this work is devoted to the properties of point spectrum which are specific to slab geometry with finite thickness.