Abstract: In algebraic topology, an Eilenberg-MacLane space is a path-connected topological space with possibly a single non-trivial homotopy group. Classifying all smooth complex algebraic Eilenberg-MacLane varieties is an interesting yet seemingly challenging question. In collaboration with R. V. Gurjar and S. R. Gurjar, we proved a classification theorem for smooth non-contractible complex affine Eilenberg-MacLane surfaces of (log) non-general type. Additionally, we proved a result about the universal cover of smooth complex projective Eilenberg-MacLane surfaces assuming an affirmative answer to an old conjecture of Igor R. Shafarevich. I will display these results in my talk. A classification theorem for smooth complex projective Eilenberg-MacLane surfaces of non-general type has been recently found by S. R. Gurjar and P. Pokale which I will mention in my talk for the sake of completeness of the above classification question for algebraic surfaces.