A survey of the homotopy nilpotency and co-nilpotency.

We review known and state some new results on the homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of the homotopy nilpotency of homogeneous spaces G/K for a Lie group G and its closed subgroup K ă G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)), Ω(Fn;n1,...,nk (K)), and Ω(Vn,m(K)) of Grassmann Gn,m(K), flag Fn;n1,...,nk (K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is studied. [ABSTRACT FROM AUTHOR]