The recently introduced rewriting calculus permits to express in a uniform and simple way first-order rewriting, lambda-calculus and non-deterministic computations as well as their combination. In this work, we recall the main properties of the rewriting calculus and we give a full first-order presentation of this rewriting calculus using an explicit substitution setting that generalizes the lambda-sigma-calculus. Its basic properties in the untyped as well as typed cases are presented. We then detail how to use the rewriting calculus to give an operational semantics to the rewrite based language ELAN.