  The conchoid of a surface $F$ with respect to given fixed point $O$ is
roughly speaking the surface obtained by increasing the radius function with
respect to $O$ by a constant. This paper studies {\it conchoid surfaces of
spheres} and shows that these surfaces admit rational parameterizations.
Explicit parameterizations of these surfaces are constructed using the
relations to pencils of quadrics in $\R^3$ and $\R^4$. Moreover we point to
remarkable geometric properties of these surfaces and their construction.
