This talk will focus on the topology of non-singular complex
algebraic surfaces, both affine and projective. Closely related is the
topology of the universal of these spaces. In many situations, this
universal cover turns out to be either an Eilenberg-MacLane space or a
Moore space.  I will discuss these properties and some related
classification theorems. The results presented will summarize joint works
with R.V. Gurjar, Buddhadev Hajra and Poonam Pokale.