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Per the European Central Bank:
TARGET2 is the real-time gross settlement (RTGS) system owned and operated by the Eurosystem. TARGET stands for Trans-European Automated Real-time Gross settlement Express Transfer system.
The 2 indicates that this is the second generation of the system.
Capital Economics shows below how TARGET2 settles and clears cross-border transactions within the eurozone. During the euro crises, significant imbalances have been created whereby the core countries have built up sizeable claims against the ECB, while peripheral countries have grown their liabilities.