  We calculate the non-abelian R-charges of BPS monopole operators in
three-dimensional gauge theories with N=3 supersymmetry. This class of models
includes ABJM theory, the proposed gauge theory dual of M-theory on AdS_4 x
S^7/Z_k, as a special case. In the UV limit of the N=3 theories the Yang-Mills
coupling becomes weak and the monopole operators are described by classical
backgrounds. This allows us to find their SU(2)_R charges in a one-loop
computation which by virtue of the non-renormalization of non-abelian R-charges
yields the exact result for any value of the coupling. The spectrum of SU(2)_R
charges is found by quantizing the SU(2)/U(1) collective coordinate of the BPS
background, whose dynamics is that of a charged particle on a sphere with a
Wess-Zumino term representing a magnetic monopole at its center. If the
Wess-Zumino coefficient is h, then the smallest possible SU(2)_R representation
for BPS monopole operators has spin |h|/2. We find, in agreement with earlier
proposals, that h is proportional to the sum of the U(1)_R charges of all the
fermion fields weighted by the effective monopole charges determined by their
gauge representations. The field content of ABJM theory is such that h=0. This
proves for any Chern-Simons level k the existence of monopole operators which
are singlets under all global symmetries and have vanishing scaling dimensions.
These operators are essential for matching the spectrum of the ABJM theory with
supergravity and for the supersymmetry enhancement to N=8.
