  Constant-curvature solutions lie at the very core of gravitational physics,
with Schwarzschild and (Anti)-de Sitter being two of the most paradigmatic
examples. Although such kind of solutions are very well-known in General
Relativity, that is not the case for theories of gravity beyond the Einsteinian
paradigm. In this article, we provide a systematic overview on $f(R)$ models
allowing for constant-curvature solutions, as well as of the constant-curvature
solutions themselves. We conclude that the vast majority of these $f(R)$ models
suffer, in general, from several shortcomings rendering their viability
extremely limited, when not ruled out by physical evidence. Among these
deficiencies are instabilities (including previously unforeseen strong-coupling
problems) and issues limiting the predictive power of the models. Furthermore,
we will also show that most $f(R)$-exclusive constant-curvature solutions also
exhibit a variety of unphysical properties.
