  In this article we introduce and study a motivic category in the arithmetic
of function fields, namely the category of motives over an algebraic closure
$L$ of a finite field with coefficients in a global function field over this
finite field. It is semi-simple, non-neutral Tannakian and possesses all the
expected fiber functors. This category generalizes the previous construction
due to Anderson and is more relevant for applications to the theory of
$G$-Shtukas, such as formulating the analog of the Langlands-Rapoport
conjecture over function fields. We further develop the analogy with the
category of motives over $L$ with coefficients in $\mathbb{Q}$ for which the
existence of the expected fiber functors depends on famous unproven
conjectures.
