  The Sombor index of a graph $G$, introduced by Ivan Gutman, is defined as the
sum of the weights $\sqrt{d_G(u)^2+d_G(v)^2}$ of all edges $uv$ of $G$, where
$d_G(u)$ denotes the degree of vertex $u$ in $G$. The Sombor coindex is
recently defined as $\bar{SO}(G)=\sum \limits_{uv\notin
E(G)}\sqrt{d_G(u)^2+d_G(v)^2}$. In this paper, the maximum and second maximum
Sombor index, the minimum and second minimum Sombor coindex in two-trees are
determined.
