  This survey paper investigates, from a purely geometric point of view,
Daniel's isometric conjugation between minimal and constant mean curvature
surfaces immersed in homogeneous Riemannian three-manifolds with isometry group
of dimension four. On the one hand, we collect the results and strategies in
the literature that have been developed so far to deal with the analysis of
conjugate surfaces and their embeddedness. On the other hand, we revisit some
constructions of constant mean curvature surfaces in the homogeneous product
spaces $\mathbb{S}^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$ and
$\mathbb{R}^3$ having different topologies and geometric properties depending
on the value of the mean curvature. Finally, we also provide some numerical
pictures using Surface Evolver.
