Justice is a contested concept. There are many different and competing conceptions, i.e. interpretations of the concept. Different domains of justice deal with different fields of application of justice claims, such as structural justice, distributive justice, participatory justice or recognition. We present a formal conceptual structure of justice applicable to all these domains. We show that conceptions of justice can be described by specifying the following conceptual elements: the judicandum (that which is to be judged as just or unjust), the community of justice including claim holders and claim addressees, their claims (and obligations), the informational base for the assessment, the principles of justice, and on a more practical level, the instruments of justice. By specifying these conceptual elements of justice, it is possible to analyse and compare different conceptions of justice, to assess their internal consistency, to explore new definitions of justice in an analytical way, and to explicate an idea of justice in a manner that provides concrete links to the relevant context.