  The notion of covering type was recently introduced by Karoubi and Weibel to
measure the complexity of a topological space by means of good coverings. When
X has the homotopy type of a finite CW-complex, its covering type coincides
with the minimum possible number of vertices of a simplicial complex homotopy
equivalent to X. In this article we compute the covering type of all closed
surfaces. Our results completely settle a problem posed by Karoubi and Weibel,
and shed more light on the relationship between the topology of surfaces and
the number of vertices of minimal triangulations from a homotopy point of view.
