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In this paper, taking into account the effect of link delays, we investigate the capacity region of the cognitive interference channel (C-IFC), where cognition can be obtained from either causal or noncausal generalized feedback. For this purpose, we introduce the causal C-IFC with delay (CC-IFC-WD) in which the cognitive user's transmission can depend on L future received symbols as well as the past ones. We show that the CC-IFC-WD model is equivalent to a classical causal C-IFC (CC-IFC) with link delays. Moreover, CC-IFC-WD extends both genie-aided and causal cognitive radio channels and bridges the gap between them. First, we derive an outer bound on the capacity region for the arbitrary value of L and specialize this general outer bound to the strong interference case. Then, under strong interference conditions, we tighten the outer bound. To derive the achievable rate regions, we concentrate on three special cases: 1) classical CC-IFC (L=0); 2) CC-IFC without delay (L=1) ; and 3) CC-IFC with unlimited look ahead in which the cognitive user noncausally knows its entire received sequence. In each case, we obtain a new inner bound on the capacity region. The derived achievable rate regions under special conditions reduce to several previously known results. Moreover, we show that the coding strategy which we use to derive an achievable rate region for the classical CC-IFC achieves the capacity for the classes of degraded and semideterministic classical CC-IFC under strong interference conditions. Furthermore, we extend our achievable rate regions to the Gaussian case. Providing some numerical examples for Gaussian CC-IFC-WD, we compare the performances of the different strategies and investigate the rate gain of the cognitive link for different delay values. We show that one can achieve larger rate regions in the “without delay” and “unlimited look-ahead” cases than in the classical CC-IFC; this improve- ent is likely due to the fact that, in the former two cases, the cognitive user can cooperate more effectively with the primary user by knowing the current and future received symbols.