In this paper we generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay’s conjecture connecting definably compact definable groups with Lie groups. Date: June 24, 2011. 2000 Mathematics Subject Classification. 03C64; 20E99. The first author was supported by Fundacao para a Ciencia e a Tecnologia, Financiamento Base 2008 ISFL/1/209. This work is part of the FCT project PTDC/MAT/101740/2008.