A contact metric (kappa, mu)-space is a contact Riemannian manifold such that the Reeb vector field belongs to the so-called (k,mu)-distribution. This new class of Riemannian manifolds was introduced by Blair, Koufogiorgos and Papantoniou in 1995 and then it was studied by many others. In this talk we shall point out the strong interplays between the theory of contact (kappa,mu)-manifolds and other topics, such as paracontact geometry or the theory of Legendre foliations. In particolar, we will show that any (k,mu)-manifold is canonically endowed with a so-called bi-paracontact structure and by using such geometric structure we will prove that any (k,mu)-space whose Boeckx invariant is different form 1 or -1 admits a compatible Sasakian or para-Sasakian metric.