Generalized Cantor manifolds and indecomposable continua

We review results concerning homogeneous compacta and discuss some open questions. It is established that indecomposable continua are Alexandroff (resp., Mazurkiewicz, or strong Cantor) manifolds with respect to the class of all continua. We also provide some new proofs of Bing's theorems about separating metric compacta by hereditarily indecomposable compacta.