  We establish a characterization of amenability for general Hausdorff
topological groups in terms of matchings with respect to finite uniform
coverings. Furthermore, we prove that it suffices to just consider two-element
uniform coverings. We also show that extremely amenable as well as compactly
approximable topological groups satisfy a perfect matching property condition
-- the latter even with regard to arbitrary uniform coverings. Finally, we
prove that the automorphism group of a Fra\"iss\'e limit of a metric
Fra\"iss\'e class is amenable if and only if the considered metric Fra\"iss\'e
class has a certain Ramsey-type matching property.
