This article is devoted to investigating zeros and fixed points of differences of entire and meromorphic functions in the complex plane, extending previous results due to Bergweiler and Langley [

*W. Bergweiler* and

*J. K. Langley*, Math. Proc. Camb. Philos. Soc. 142, No. 1, 133–147 (2007;

Zbl 1114.30028)]. Denoting, for a transcendental meromorphic function

$f$,

${\Delta}f\left(z\right):=f(z+c)-f\left(z\right)$,

${{\Delta}}^{n+1}f\left(z\right):={{\Delta}}^{n}f(z+c)-{{\Delta}}^{n}f\left(z\right)$, the results obtained are treating differences and divided differences

$G\left(z\right):={\Delta}f\left(z\right)/f\left(z\right)$,

${G}_{n}\left(z\right):={{\Delta}}^{n}\left(z\right)/f\left(z\right)$ of

$f$. The key results in this paper are as follows: (1)

${G}_{n}\left(z\right)$ has infinitely many zeros and infinitely many fixed points, provided

$f$ is entire,

$c=1$ and

$\rho \left(f\right)<1/2$ and

$\rho \left(f\right)\ne j/n$,

$j=1,...,[n/2]$. (2)

$G\left(z\right)$ has infinitely many zeros and infinitely many fixed points, whenever

$f$ is entire with

$\rho \left(f\right)<1$, and either

$f$ has at most finitely many zeros whose difference is

$=c$ or

${lim\; inf}_{j\to \infty}|{z}_{j+1}/{z}_{j}|=L>1$, where

$\left\{{z}_{j}\right\}$ is the zero-sequence of

$f$, arranged according to increasing moduli. (3) A similar result holds, if

$f$ is entire of order

$\rho \left(f\right)=1$ and with infinitely many zeros having the exponent of convergence

$\lambda \left(f\right)<1$. (4) As for the case of

$f$ meromorphic, a result similar to (2) follows by invoking corresponding conditions for the poles of

$f$ as well. (5) Given a positive, non-decreasing function

$\phi :[1,\infty )\to [0,\infty )$ with

${lim}_{r\to \infty}\phi \left(r\right)=\infty $, there exists

$f$ transcendental meromorphic such that

${lim\; sup}_{r\to \infty}(T(r,f)/r)<\infty $,

${lim\; sup}_{r\to \infty}(T(r,f)/\phi \left(r\right)logr)<\infty $ and that

${\Delta}f\left(z\right)$ has one fixed point only. The proofs rely on standard properties of meromorphic functions, Wiman-Valiron theory and some Nevanlinna theory.