The $\rho$-calculus is a new calculus that integrates in a uniform and simple setting first-order rewriting, $\lambda$-calculus and non-deterministic computations. This paper describes the calculus from its syntax to its basic properties in the untyped case. We show how it embeds first-order conditional rewriting and $\lambda$-calculus. Finally we use the $\rho$-calcul- us to give an operational semantics to the rewrite based language ELAN.