  In this paper, we investigate the geometry and classification of
three-dimensional CR Yamabe solitons. In the compact case, we show that any
3-dimensional CR Yamabe soliton must have constant Tanaka-Webster scalar
curvature; we also obtain a classification under the assumption that their
potential functions are in the kernel of the CR Paneitz operator. In the
complete case, we obtain a structure theorem on the diffeomorphism types of
complete 3-dimensional pseudo-gradient CR Yamabe solitons (shrinking, or
steady, or expanding) of vanishing torsion.
