  We study skew-amenable topological groups, i.e., those admitting a
left-invariant mean on the space of bounded real-valued functions
left-uniformly continuous in the sense of Bourbaki. We prove characterizations
of skew-amenability for topological groups of isometries and automorphisms,
clarify the connection with extensive amenability of group actions, establish a
F{\o}lner-type characterization, and discuss closure properties of the class of
skew-amenable topological groups. Moreover, we isolate a dynamical sufficient
condition for skew-amenability and provide several concrete variations of this
criterion in the context of transformation groups. These results are then used
to decide skew-amenability for a number of examples of topological groups built
from or related to Thompson's group $F$ and Monod's group of piecewise
projective homeomorphisms of the real line.
