Summary: This paper is devoted to the attraction-repulsion chemotaxis system with a logistic source: \[ \begin{cases} u_t= \Delta u- \chi \nabla (u \nabla v)+ \mu \nabla (u \nabla w) + \mathcal{R}(u), \; x\in \Omega,\;t>0,\\ v_t= \Delta v- \alpha_1 v + \beta_1 u, \; x\in \Omega,\;t>0\\ w_t=\Delta w-\alpha_2 w + \beta_2 u, \; x\in \Omega,\;t>0 \end{cases} \]
where \(\Omega \subset \mathbb {R}^N(N\geqslant 1)\) is a bounded domain with smooth boundary and \(\mathcal {R}(s)\leqslant a-bs^\tau \). For the case \(\varrho =0\), we show that when the repulsion prevails over the attraction in the sense that \(\mu \beta _2-\chi \beta _1>0\), there exist global bounded classical solutions for any logistic damping \(\tau \geqslant 1\). When the attraction dominates the repulsion in the sense that \(\mu \beta _2-\chi \beta _1<0\), the classical solutions are still global and bounded provided that the logistic damping is strong. For the case \(\varrho >0\), we will investigate the similar problem for \(N=1\) and \(N=2\). We will also study the regularity of stationary solutions.