Warsaw University Faculty of Mathematics , Informatics and Mechanics Karol Palka

The thesis is devoted to studying normal complex Q-acyclic algebraic surfaces S ′. Let S0 be the smooth locus of such a surface. The following results have been obtained. If S ′ has non-negative Kodaira dimension then it is logarithmic, i.e. its singularities are of quotient type. We classify possible S ′ with non-quotient singularities. S ′ can be nonrational. The completion of the resolution of S ′ is birationally ruled, i.e. it is a a blowup of a Pbundle over some smooth complete curve. We classify possible S ′ for which κ(S0) = 0 and S0 does not admit a C∗-fibration. The main result is the theorem saying that if κ(S ′) = −∞ then κ(S0) 6= 2. The full description of possible singular Q-homology planes S ′ of negative Kodaira dimension is given.