• compact domination
• definable groups
• NIP
• o-minimality
• Pillay's conjecture

In 2004 Pillay conjectured that if G is a definable compact group in a sufficiently saturated o-minimal structure then:
1) G admits a minimal type-definable normal subgroup of bounded index, call it G00.
2) G/G00, equipped with the logic topology, is a compact real Lie group of the same dimension as G.
The beauty of this statement is that it offers a surprising connection between the pure lattice of definable sets in definable groups in o-minimal structures and real Lie groups.

However, when a proof of the full conjecture was finally found, it was somewhat unsatisfactory from a model theoretic point of view, as it made use of several external tools coming from algebraic topology and other subjects. Nonetheless, it helped to introduce the use of measures in model theory, and made clear that the right framework in which to study these objects was the more general setting of NIP theories.
These intuitions recently developed in a new powerful theory of groups and measures in the NIP settings.

The aim of this work is therefore to give a concise and self-contained exposition of the theory of groups definable in o-minimal structures as seen from the point of view of NIP theories. A great deal of attention has been paid in using the recent advancements in the subject to eliminate any unnecessary prerequisite from the original proofs, and to rewrite them using the most recent model theoretic viewpoint.

Among the other results, we describe a relatively short proof of the compact domination property for definably compact groups.