  We study conformal metrics on $R^3$, i.e., metrics of the form
$g_u=e^{2u}|dx|^2$, which have constant $Q$-curvature and finite volume. This
is equivalent to studying the non-local equation $$ (-\Delta)^\frac32 u = 2
e^{3u}$$ in $R^3$ $$V:=\int_{\mathbb{R}^3}e^{3u}dx<\infty,$$ where $V$ is the
volume of $g_u$. Adapting a technique of A. Chang and W-X. Chen to the
non-local framework, we show the existence of a large class of such metrics,
particularly for $V\le 2\pi^2=|S^3|$. Inspired by previous works of C-S. Lin
and L. Martinazzi, who treated the analogue cases in even dimensions, we
classify such metrics based on their behavior at infinity.
