  We give a complete classification of solutions bounded from above of the
Liouville equation $$-\Delta u=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More
generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty}
u(z)/\log|z|:=k(u)<\infty\}$$ are described. As a consequence, we obtain five
rigidity results. First, $k(u)$ can take only a discrete set of values: either
$k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as
$z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is
symmetric with respect to $x$ and $y$ axes and $u_x<0,\; u_y<0$ in the first
quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded
from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above,
and the diameter of ${\mathbf{R}}^2$ with the metric $e^{2u}\delta$ is $\pi$,
where $\delta$ is the Euclidean metric, then $u$ is either radial about a point
or one-dimensional.
  In addition, we extend the concavity rigidity result on Liouville equation in
higher dimensions.
