  We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global
function field $F$ and study various aspects of Iwasawa theory with emphasis on
the two main themes already (and still) developed in the number fields case as
well. When dealing with the Selmer group of an abelian variety $A$ defined over
$F$, we provide all the ingredients to formulate an Iwasawa Main Conjecture
relating the Fitting ideal and the $p$-adic $L$-function associated to $A$ and
$\mathcal{F}$. We do the same, with characteristic ideals and $p$-adic
$L$-functions, in the case of class groups (using known results on
characteristic ideals and Stickelberger elements for
$\mathbb{Z}_p^d$-extensions). The final section provides more details for the
cyclotomic $\mathbb{Z}_p^{\mathbb{N}}$-extension arising from the torsion of
the Carlitz module: in particular, we relate cyclotomic units with
Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism.
