  If $\textbf{S}$ is a subcategory of metric spaces, we say that a group G has
property $B\textbf{S}$ if any isometric action on an $\textbf{S}$-space has
bounded orbits. Examples of such subcategories include metric spaces, affine
real Hilbert spaces, CAT(0) cube complexes, connected median graphs, trees or
ultra-metric spaces. The corresponding properties $B\textbf{S}$ are
respectively Bergman's property, property FH (which, for countable groups, is
equivalent to the celebrated Kazhdan's property (T)), property FW (both for
CAT(0) cube complexes and for connected median graphs), property FA and
uncountable cofinality. Historically many of these properties were defined
using the existence of fixed points.
  Our main result is that for many subcategories $\textbf{S}$, the wreath
product $G\wr_XH$ has property $B\textbf{S}$ if and only if both $G$ and $H$
have property $B\textbf{S}$ and $X$ is finite. On one hand, this encompasses in
a general setting previously known results for properties FH and FW. On the
other hand, this also applies to the Bergman's property. Finally, we also
obtain that $G\wr_XH$ has uncountable cofinality if and only if both $G$ and
$H$ have uncountable cofinality and $H$ acts on $X$ with finitely many orbits.
