The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-) Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class Ω of all locally compact homogeneous spaces with inner metric is supplied with some metric d BGH such that 1) (Ω, d BGH) is a complete metric space; 2) a sequences in (Ω, d BGH) is

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... onverging if and only if it is converging in Gromov-Hausdor sense; 3) the subclasses M of homogeneous manifolds with inner metric and L G of connected Lie groups with left-invariant Finslerian metric are everywhere dense in (Ω, d BGH): It is given a metric characterization of Carnot groups with left-invariant sub-Finslerian metric. At the end are described homogeneous manifolds such that any invariant inner metric on any of them is Finslerian.