Abstract: Complex contact manifolds generalise real notion, which originates from classical mechanics. However in the complex case the known examples are very rare, and according to LeBrun-Salamon conjecture, there is no more projective examples. Projective contact manifolds are studied in relation with a classification of Riemannian manifolds with respect to their holonomy groups. We pose the question if there are any other examples of non-projective contact manifolds.  I will present a simple result obtained with Thomas Peternell, which proves the classification in dimension 3 and with first Betti number 1. Further I will explain how the non-projective classification or examples (in any dimension) can potentially affect the projective classification.