  This is a review of explicit computations of Connes distance in
noncommutative geometry, covering finite dimensional spectral triples,
almost-commutative geometries, and spectral triples on the algebra of compact
operators. Several applications to physics are covered, like the metric
interpretation of the Higgs field, and the comparison of Connes distance with
the minimal length that emerges in various models of quantum spacetime. Links
with other areas of mathematics are studied, in particular the horizontal
distance in sub-Riemannian geometry. The interpretation of Connes distance as a
noncommutative version of the Monge-Kantorovich metric in optimal transport is
also discussed.
