Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions, with K-contact manifolds corresponding to symplectic manifolds. It is an interesting problem to find obstructions for a closed manifold to admit such types of structures and in particular, to construct K-contact manifolds which do not admit Sasakian structures. In the simply-connected case, the hardest dimension is 5, where Kollár has found subtle obstructions to the existence of Sasakian structures, associated to the theory of algebraic surfaces. In this talk, we develop methods to distinguish K-contact manifolds from Sasakian ones in dimension 5. In particular, we find the first example of a closed 5-manifold with first Betti number \( b_1 = 0\) which is K-contact but which carries no semi-regular Sasakian structure (Joint work with J.A. Rojo and A. Tralle).