Topology of definable groups in tame structures

I will survey some results on definable groups in o-minimal structures, some old, some new, emphasizing the interplay between algebra, logic, and topology. In particular I will show how a combination of techniques from model theory and algebraic topology lead to the determination of the definable homeomorphism type of definable abelian groups in dimension not equal to 4 (joint work with E. Baro). If time permits, I will consider the problem of finding a tame definable context, larger than o-minimality, which is suitable for the study of universal covers (work in progress with M. Mamino).