Unlinkability is a privacy property of crucial importance for several systems (such as RFID or voting systems). Informally, unlinkability states that, given two events/items in a system, an attacker is not able to infer whether they are related to each other. However, in the literature we find several definitions for this notion, which are apparently unrelated and shows a potentially problematic lack of agreement. This paper sheds new light on unlinkability by comparing different ways of defining it and showing that in many practical situations the various definitions coincide. It does so by (a) expressing in a unifying framework four definitions of unlinkability from the literature (b) demonstrating how these definitions are different yet related to each other and to their dual notion of ''inseparability'' and (c) by identifying conditions under which all these definitions become equivalent. We argue that the conditions are reasonable to expect in identification systems, and we prove that they hold for a generic class of protocols.