  Global properties of vacuum static, spherically symmetric configurations are
studied in a general class of scalar-tensor theories (STT) of gravity in
various dimensions. The conformal mapping between the Jordan and Einstein
frames is used as a tool. Necessary and sufficient conditions are found for the
existence of solutions admitting a conformal continuation (CC). The latter
means that a singularity in the Einstein-frame manifold maps to a regular
surface S_(trans) in the Jordan frame, and the solution is then continued
beyond this surface. S_(trans) can be an ordinary regular sphere or a horizon.
In the second case, S_(trans) proves to connect two epochs of a Kantowski-Sachs
type cosmology. It is shown that, in an arbitrary STT, with arbitrary potential
functions $U(\phi)$, the list of possible types of causal structures of vacuum
space-times is the same as in general relativity with a cosmological constant.
This is true even for conformally continued solutions. It is found that when
S_(trans) is an ordinary sphere, one of the generic structures appearing as a
result of CC is a traversable wormhole. Two explicit examples are presented: a
known solution illustrating the emergence of singularities and wormholes, and a
nonsingular 3-dimensional model with an infinite sequence of CCs.
