Abstract The paper is dedicated to the study of the problem of the existence of compact global attractors of discrete inclusions and to the description of its structure. We consider a family of continuous mappings of a metric space W into itself, and ( W , f i ) i ∈ I is the family of discrete dynamical systems. On the metric space W we consider a discrete inclusion (1) u t + 1 ∈ F ( u t ) associated with M ≔ { f i : i ∈ I } , where F ( u ) = { f ( u ) : f ∈ M } for all u ∈ W . We give sufficient conditions (the family of maps M is contracting in the extended sense) for the existence of a compact global attractor of (1) . If the family M consists of a finite number of maps, then the corresponding compact global attractor is chaotic. We study this problem in the framework of non-autonomous dynamical systems (cocyles).