Summary: Distance-hereditary graphs have been introduced by Howorka and studied in the literature with respect to their metric properties. In this paper several equivalent characterizations of these graphs are given: in terms of existence of particular kinds of vertices (isolated, leaves, twins) and in terms of properties of connections, separators, and hangings. Distance-hereditary graphs are then studied from the algorithmic viewpoint: simple recognition algorithms are given and it is shown that the problems of finding cardinality Steiner trees and connected dominating sets are polynomially solvable in a distance-hereditary graph.