We calculate the anomalous dimensions of operators with large global charge $J$ in certain strongly coupled conformal field theories in three dimensions, such as the $O(2)$ model and the supersymmetric fixed point with a single chiral superfield and a $W = \Phi^3$ superpotential. Working in a $1/J$ expansion, we find that the large-$J$ sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge $J$ is always a scalar operator whose dimension $ \Delta(J)$ satisfies the sum rule $ J^2 \Delta(J) - ( \frac{J^2}{2} + \frac{J}{4} + \frac{3}{16}) \Delta(J-1) - ( \frac{J^2}{2} - \frac{J}{4} + \frac{3}{16} ) \Delta(J+1) = 0.04067 $ up to corrections that vanish at large $J$. The spectrum of low-lying excited states is also calculable explcitly: for example, the second-lowest primary operator has spin two and dimension $\Delta(J) + \sqrt{3}$. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order $ J^{1 / 2} $. The propagation speeds of the Goldstone waves and heavy fermions are $1 / \sqrt{2}$ and $\pm 1 / 2$ times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large $J$.