There are three Bieberbach theorems on flat Riemannian manifolds; characterization, rigidity and finiteness. These extend to almost flat manifolds. We discuss characterization, rigidity and finiteness of infra-nilmanifolds (almost flat manifolds). We apply these to construction of aspherical spaces, calculation of fixed point theory invariants. We will start with 3-dimensional manifolds, their fiber structures. Also we discuss some specific examples of nilpotent Lie groups.