Abstract quotients of profinite groups, after Nikolov and Segal

In this expanded account of a talk given at the Oberwolfach Arbeitsgemeinschaft "Totally Disconnected Groups", October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is a paper, entitled "Generators and commutators in finite groups; abstract quotients of compact groups", that was published by Nikolov and Segal in 2012. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject.