Some classes of finite differences that preserve roots of univariate polynomials on lines or in strips and half-planes of the complex plane will be presented. In particular, we describe some classes of finite differences that preserve the hyperbolicity (real-rootedness) of polynomials and prove a finite difference analogue of the Hermite-Pauline theorem (completely different from the one recently established by Br/"and/'en, Krasikov and Shapiro).We also found the polynomial whose finite differences has the minimal mesh (minimal distance between roots) among all other polynomials. Corresponding results for entire functions will be presented. Finally, some asymptotic results for roots of finite differences of polynomials will be presented.