  The Haagerup property for locally compact groups is generalised to the
context of locally compact quantum groups, with several equivalent
characterisations in terms of the unitary representations and positive-definite
functions established. In particular it is shown that a locally compact quantum
group G has the Haagerup property if and only if its mixing representations are
dense in the space of all unitary representations. For discrete G we
characterise the Haagerup property by the existence of a symmetric proper
conditionally negative functional on the dual quantum group $\hat{G}$; by the
existence of a real proper cocycle on G, and further, if G is also unimodular
we show that the Haagerup property is a von Neumann property of G. This extends
results of Akemann, Walter, Bekka, Cherix, Valette, and Jolissaint to the
quantum setting and provides a connection to the recent work of Brannan. We use
these characterisations to show that the Haagerup property is preserved under
free products of discrete quantum groups.
