  We give an accessible introduction into the theory of lower central series of
associative algebras, exhibiting the interplay between algebra, geometry and
representation theory that is characteristic for this subject, and to discuss
some open questions. In particular, we provide shorter and clearer proofs of
the main results of this theory. We also discuss some new theoretical and
computational results and conjectures on the lower central series of the free
algebra in two generators modulo a generic homogeneous relation.
