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We define a metric ultraproduct of topological groups with left-invariantmetric, and show that there is a countable sequence of finite groups withleft-invariant metric whose metric ultraproduct contains isometrically as asubgroup every separable topological group with left-invariant metric.  In particular, there is a countable sequence of finite groups withleft-invariant metric such that every finite subset of an arbitrary topologicalgroup with left-invariant metric may be approximated by all but finitely manyof them.  We compare our results with related concepts such as sofic groups,hyperlinear groups and weakly sofic groups.

Metric topological groups: their metric approximation and metric ultraproducts