In 1841 Delaunay characterized surfaces of constant mean curvature $H=1$ in Euclidean 3-space invariant under rotation. The result was generalized by several authors to screw-motion invariant CMC surfaces in $\mathbb{E}(\kappa,\tau)$. In this more general setting CMC tubes can arise in addition to the Delaunay surfaces. In this talk I want to present existence conditions and talk about further properties of these tubes such as embeddedness and foliation.