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Power transformers are important and expensive components in the electric power system. The de-regulation of electric power requires a reduction of the service and maintenance cost of the power utilities. Monitoring systems can help to decrease the transformer life cycle cost and to increase the high level of availability and reliability. On line monitoring is the record of relevant data of a transformer. Diagnosis is the interpretation of these monitored data including the history of the transformer and the statistical judgement of the failure rate. The importance of the monitored transformer and the economic consequences are the basis for the asset management of power transformer together with the risk evaluation. This paper attempts to present the different methodologies adopted for on line monitoring of power transformers.