The rewriting calculus is a new calculus that integrates in a uniform and simple setting first-order rewriting, lambda-calculus and non-deterministic computations. The main design concept of the rewriting calculus is to make all the basic ingredients of rewriting explicit objects, in particular the notions of rule application and multisets of results. This paper describes the calculus from its syntax to its basic properties in the untyped case. The rewriting calculus embeds first-order conditional rewriting and lambda-calculus and it can be used in order to give an operational semantics to the rewrite based language Elan. We show how the set-like data structures are easily represented in Elan and how this can be used in order to specify protocols.