Are there metric theories of gravity other than general relativity?

Some well-known generalizations of general relativity are analyzed. According to the author, two gravitational theories are said to be physically equivalent if: (a) the gravitational field has the same vacuum dynamics; and (b) the coupling between the gravitational metric (or the gravitational connection) and any kind of matter is the same for both theories. The cases considered correspond to: purely affine (gravitational field described by a symmetric linear connection), metric-affine (metric and connection considered as independent variables in the variational principle), scalar-tensor and higher-order derivative theories. It is argued that, as far as criterion (a) is concerned, the theories considered are physically equivalent, up to some field redefinitions, to general relativity.