Antiunitary operators are characterized in a manner similar to the characterization of unitary operators by their characteristic vectors and characteristic values. It is shown that a complete orthonormal set of vectors can be defined, some of which are invariant under the antiunitary operator. The rest of the vectors, which are always even in number, form pairs in such a way that the antiunitary operator transforms each member of a pair into a multiple of the other member of the same pair [Eq. (11)]. The extent to which the vectors of the orthonormal set are determined by the antiunitary operator is ascertained and the number of free parameters in the various cases of degeneracy found.