  We study static spherically symmetric solutions of Einstein gravity plus an
action polynomial in the Ricci scalar, $R$, of arbitrary degree, $n$, in
arbitrary dimension, $D$. The global properties of all such solutions are
derived by studying the phase space of field equations in the equivalent theory
of gravity coupled to a scalar field, which is obtained by a field redefinition
and conformal transformation. The following uniqueness theorem is obtained:
provided that the coefficient of the $R^2$ term in the Lagrangian polynomial is
positive then the only static spherically symmetric asymptotically flat
solution with a regular horizon in these models is the Schwarzschild solution.
Other branches of solutions with regular horizons, which are asymptotically
anti-de Sitter, or de Sitter, are also found. An exact Schwarzschild-de Sitter
type solution is found to exist in the $R+aR^2$ if $D>4$. If terms of cubic or
higher order in $R$ are included in the action, then such solutions also exist
in four dimensions. The general Schwarzschild-de Sitter type solution for
arbitrary $D$ and $n$ is given. The fact that the Schwarzschild solution in
these models does not coincide with the exterior solution of physical bodies
such as stars has important physical implications which we discuss. As a
byproduct, we classify all static spherically symmetric solutions of
$D$-dimensional gravity coupled to a scalar field with a potential consisting
of a finite sum of exponential terms.
