  We present a comprehensive theory of boundedness properties for Polish groups
developed with a main focus on Roelcke precompactness (precompactness of the
lower uniformity) and Property (OB) (boundedness of all isometric actions on
separable metric spaces). In particular, these properties are characterised by
the orbit structure of isometric actions on metric spaces and isometric or
continuous affine representations on separable Banach spaces.
