  We introduce the weak Haagerup property for locally compact groups and prove
several hereditary results for the class of groups with this approximation
property. The class contains a priori all weakly amenable groups and groups
with the (usual) Haagerup property, but examples are given of groups with the
weak Haagerup property which are not weakly amenable and do not have the
Haagerup property.
  In the second part of the paper we introduce the weak Haagerup property for
finite von Neumann algebras, and we prove several hereditary results here as
well. Also, a discrete group has the weak Haagerup property if and only if its
group von Neumann algebra does.
  Finally, we give an example of two II_1 factors with different weak Haagerup
constants.
