  We study hypersurfaces of the four-dimensional Thurston geometry
$\text{Sol}^4_0$, which is a Riemannian homogeneous space and a solvable Lie
group. In particular, we give a full classification of hypersurfaces whose
second fundamental form is a Codazzi tensor, including totally geodesic
hypersurfaces and hypersurfaces with parallel second fundamental form, and of
totally umbilical hypersurfaces of $\text{Sol}^4_0$. We also give a closed
expression for the Riemann curvature tensor of $\text{Sol}^4_0$, using two
integrable complex structures.
