[For the entire collection see Zbl 0651.00007.]
This is a survey of results on representation varieties of Lie groups and algebras. Finite groups, finite-dimensional algebras, commutative algebras, finite-dimensional Lie algebras, finitely generated nilpotent groups, fundamental groups of compact, orientable surfaces are considered by examples. The problem consists in the characterization of connected components of representation varieties starting from the structure of the group \(\Gamma\) or the algebra A, or from the topological structure of the surface (for the fundamental group of the surface). For instance, for the fundamental group \(\pi_ 1(M)\) of a compact, orientable surface M of genus \(g\geq 2\), the variety of representations \(R(\pi_ 1(M),SL_ 2({\mathbb{R}}))\) consists of \(2^{2g+1}+2g-3\) components. It is known that \(R(\pi_ 1(M),GL_ 2({\mathbb{C}}))\) is connected if the genus g(M)\(\geq 0\) and \(R(\pi_ 1(M),GL_ n({\mathbb{C}}))\) is also connected, if \(g(M)=0\) or 1. The question of the connectedness of \(R(\pi_ 1(M),GL_ n({\mathbb{C}}))\) in the case g(M)\(\geq 2\) or \(n\geq 3\) remains open.