  In this paper we construct Galois towers with good asymptotic properties over
any non-prime finite field $\mathbb F_{\ell}$; i.e., we construct sequences of
function fields $\mathcal{N}=(N_1 \subset N_2 \subset \cdots)$ over $\mathbb
F_{\ell}$ of increasing genus, such that all the extensions $N_i/N_1$ are
Galois extensions and the number of rational places of these function fields
grows linearly with the genus. The limits of the towers satisfy the same lower
bounds as the best currently known lower bounds for the Ihara constant for
non-prime finite fields. Towers with these properties are important for
applications in various fields including coding theory and cryptography.
