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We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a globalfunction field $F$ and study various aspects of Iwasawa theory with emphasis onthe two main themes already (and still) developed in the number fields case aswell. When dealing with the Selmer group of an abelian variety $A$ defined over$F$, we provide all the ingredients to formulate an Iwasawa Main Conjecturerelating 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 oncharacteristic ideals and Stickelberger elements for$\mathbb{Z}_p^d$-extensions). The final section provides more details for thecyclotomic $\mathbb{Z}_p^{\mathbb{N}}$-extension arising from the torsion ofthe Carlitz module: in particular, we relate cyclotomic units withBernoulli-Carlitz numbers by a Coates-Wiles homomorphism.

Aspects of Iwasawa theory over function fields