Abstract: In the talk I will present recent results obtained in collaboration with J. Alt and A. J. Di Scala about holonomy reductions of normal conformal Cartan connections. Using a recent result by Cap, Gover and Hammerl, we study the case when the conformal holonomy reduces to the isotropy group H of a pseudo-Riemannian irreducible symmetric space G/H, and find that, under certain algebraic conditions, such a reduction not only specifies a metric g in the conformal class but also a metric connection D with totally skew symmetric torsion and holonomy in the stabiliser in H of a null line. Special cases of this situation are the symmetric spaces defined by G=SL(n,R) and H=SO(p,q), in which case we show that the metric g is flat and hence that the conformal class must be flat. Further examples are the symmetric spaces G=SL(3,K), where K are the complex numbers or the quaternions and H is either SU(1,2) or Sp(1,2). It turns out that in both cases the metric g must be Einstein, and moreover, in the complex case that D defines a nearly-para-Kaehler structure. As a consequence we can exclude these H as possible conformal holonomy groups.