In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel $T_{\Omega}$ on the generalized local Morrey spaces $LM_{p,\varphi}^{\{x_{0}\}}$ , for $s' \le p$ , $p \neq1$ or $p < s$ , where $\Omega\in L_{s}(S^{n-1})$ with $s>1$ are homogeneous of degree zero. In the case when $b \in LC_{p,\lambda}^{\{x_{0}\}}$ is a local Campanato spaces, $1< p<\infty$ , and $T_{\Omega,b}$ be is a sublinear commutator operator, we find the sufficient conditions on the pair $(\varphi_{1},\varphi_{2})$ which ensures the boundedness of the operator $T_{\Omega,b}$ from one generalized local Morrey space $LM_{p,\varphi_{1}}^{\{x_{0}\}}$ to another $LM_{p,\varphi_{2}}^{\{x_{0}\}}$ . In all cases the conditions for the boundedness of $T_{\Omega}$ are given in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$ , which do not make any assumptions on the monotonicity of $\varphi_{1}$ , $\varphi_{2}$ in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudo-differential operators, Littlewood-Paley operators, Marcinkiewicz operators, and Bochner-Riesz operators.