A countable discrete group $G$ is $C^\ast$-exact or simply, exact, if its reduced $C^\ast$-algebra $C^\ast_r(G)$ is an exact $C^\ast$-algebra, i.e. if taking the minimal tensor product with $C_r^\ast(G)$ preserves short exact sequences of $C^\ast$-algebras. The exactness is viewed as a weak amenability. All amenable groups, linear groups, Gromov?s hyperbolic groups, groups with finite asymptotic dimension, and many other familiar groups are known to be exact. In contrast, constructions of non-exact groups are rare and technically quite involved. We will discuss such constructions, indicate applications, and suggest some open problems.