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In this paper, we investigate the existence of a competitive equilibrium in an intertemporal pure exchange model with a countable number of periods and a countable number of agents. This model covers both the overlapping generations case and the case where there is an infinite number of infinite-lived agents or perhaps a mixture of both infinite-lived and finite-lived agents. At each period, the commodity space can be finite or infinite dimensional. A first equilibrium existence theorem is established for this general model under the classical assumption that there exists a finite set of non-negligible agents. Applied to the overlapping generations exchange model, this result implies for the agents the possibility of having endowments outside their life-time. The existence of an equilibrium for the overlapping generations model when the agents have no endowment outside their life-time is then proved. The last section extends this result to the case of commodity spaces without order-unit element.