We discuss the global properties of static, spherically symmetric configurations of a self-gravitating real scalar field phi in General Relativity (GR), scalar-tensor and high-order (curvature-nonlinear) theories of gravity in various dimensions. In GR, for scalar fields with an arbitrary potential V(phi), not necessarily positive-definite, it is shown that the list of all possible types of space-time causal structure in the models under consideration is the same as the one for phi = const. In particular, there are no regular black holes with any asymptotics. These features are extended to scalar-tensor and curvature-nonlinear gravity, connected with GR by conformal mappings, unless there is a conformal continuation, i.e., a case when a singularity in a solution of GR maps to a regular surface in an alternative theory, and the solution is continued through such a surface. Such an effect is exemplified by exact solutions in GR with a massless conformal scalar field, considered as a special scalar-tensor theory. Necessary conditions are found for the existence of a conformal continuation; they only hold for special choices of scalar-tensor and high-order theories of gravity.