Non-commutative Iwasawa theory has emerged as a powerful framework for understanding deep arithmetic properties over number fields contained in a p-adic Lie extension and their precise relationship to special values of complex L-functions. This talk aims to explore non-commutative Iwasawa theory over global function fields. We consider an abelian variety A defined over various base fields F, and discuss its arithmetic over the cyclotomic Z_p-extension and more general p-adic Lie extensions. After reviewing some known results over number fields, we shift our focus to the case of global function fields. In this context, we compare the arithmetic of A over different p-adic Lie extensions without assuming the finiteness of the Selmer group of A over the base field F. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.