In this paper we give a new simpler and uniform presentation of the rewriting calculus also called Rho Calculus. In addition to its simplicity, this reformulation explicitly allows us to encode complex structures such as lists, sets, and objects. We provide extensive examples of calculus use and we focus on its properties and its ability to represent some object oriented calculi, namely the Lambda Calculus of Objects of Fisher, Honsell, and Mitchell, and the Object Calculus of Abadi and Cardelli. This enlightens the capabilities of the rewriting calculus based language ELAN to be used as a logical as well as powerful semantical framework. In summa, we intend to show that the Rho Calculus represents a lingua franca to encode many paradigms of computations.