  The nonnegativity of the CR Paneitz operator plays a crucial role in
three-dimensional CR geometry. In this paper, we prove this nonnegativity for
embeddable CR manifolds. This result and previous works give an affirmative
solution of the CR Yamabe problem for embeddable CR manifolds. We also show the
existence of a contact form with zero CR $Q$-curvature, and generalize the
total $Q$-prime curvature to embeddable CR manifolds with no pseudo-Einstein
contact forms. Furthermore, we discuss the logarithmic singularity of the
Szeg\H{o} kernel.

Differential Geometry;Analysis of PDEs;Complex Variables