The aim of this series of talks is to give an outline of the speaker's research on various topics in group theory and categorical algebra. Many of the new tools and approaches introduced actually originate from former work in unstable homotopy theory, in particular appropriate notions of polynomial maps between algebraic objects and of polynomial functors. The latter is part of a calculus for functors initiated by Eilenberg and MacLane and later further developed and extended in several directions by numerous authors, in particular to a homotopical setting by T. Goodwillie and to non-abelian algebraic settings first by Baues and Pirashvili and later by the speaker and collaborators.