  We consider two types of minimal Poincar\'e $4$-complexes. One is defined
with respect to the degree $1$-map order. This idea was already present in our
previous papers, and more systematically studied later by Hillman. The second
type of minimal Poincar\'e $4$-complexes were introduced by Hambleton, Kreck
and Teichner. It is not based on an order relation. In the present paper we
study existence and uniqueness.
