  Let $X$ be a simply connected space and ${\Bbb F}_p$ be a prime field. The
algebra of normalized singular cochains $N^*(X; {\Bbb F}_p)$ admits a natural
homotopy structure which induces natural Steenrod operations on the Hochschild
homology $HH_* N^*(X;{\Bbb F}_p)$ of the space $X$. The primary purpose of this
paper is to prove that the J. Jones isomorphism $HH_*N^*(X;{\Bbb F}_p) \cong H
^*(X^{S^1};{\Bbb F}_p)$ identifies theses Stenrood operations with those
defined on the cohomology of the free loop space with coefficients in ${\Bbb
F}_p$. The other goal of this paper is to describe a theoritic model which
allows to do some computations.
