We introduce problems of decentralized control with delayed communication, where delays are either unbounded or bounded by a given constant k. In the k-bounded-delay model, between the transmission of a message and its reception, the plant can execute at most k events. In the unbounded-delay model, the plant can execute any number of events between transmission and reception. We show that our framework yields an infinite hierarchy of control problems, CC = DCC0 superset of DCC1 superset of DCC2 superset of (...) superset of DCUC superset of DC where the containments are strict, CC is the set of control problems solvable with a single controller (centralized case) and DCCk (respectively, DCUC,DC) is the set of problems solvable with two controllers in a k-bounded-delay network (respectively, in an unbounded-delay network, without communication). The hierarchy is a result of the following property: Controllers which "work" in a given network will also work in a less nondeterministic network. This property. does not hold when nonblockingness is introduced. Checking the existence of controllers in the unbounded-delay case or in the case without communication are undecidable problems. However, a related decentralized observation problem with bounded-delay communication is decidable.