Choose $q\in ℂ$ with $0\<\left|q\right|\<1$. The main theme of this paper is the study of linear $q$-difference equations over the field $K$ of germs of meromorphic functions at $0$. A systematic treatment of classification and moduli is developed. It turns out that a difference module $M$ over $K$ induces in a functorial way a vector bundle $v\left(M\right)$ on the Tate curve $E_q:={ℂ}^{*}/q^{\mathbb{Z}}$ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification $\left(\right[\mathrm{A}t\left]\right)$ of the indecomposable vector bundles on the complex Tate...