  We prove some theorems characterizing the global properties of static,
spherically symmetric configurations of a self-gravitating real scalar field in
general relativity (GR) in various dimensions, with an arbitrary potential $V$,
not necessarily positive-definite. The results are extended to sigma models,
scalar-tensor and curvature-nonlinear theories of gravity. We show that the
list of all possible types of space-time causal structure in the models under
study is the same as for a constant scalar field, namely, Minkowski (or AdS),
Schwarzschild, de Sitter and Schwarzschild - de Sitter, and all horizons are
simple. In particular, these theories do not admit regular black holes with any
asymptotics. Some special features of (2+1)D gravity are revealed. We give
examples of two types of asymptotically flat configurations with positive mass
in GR, admitted by the above theorems: (i) a black hole with nontrivial
``scalar hair'' and (ii) a particlelike solution with a regular centre; in both
cases, the potential $V$ must be at least partly negative. We also discuss the
global effects of conformal mappings that connect different theories. Such
effects are illustrated for solutions with a conformal scalar field in GR.
