In this paper, we consider a semi-infinite multiobjective optimization problem with more than two differentiable objective functions and uncertain constraint functions, which is called a robust semi-infinite multiobjective optimization problem and give its robust counterpart \({\mathrm{(RSIMP)}}\) of the problem, which is regarded as the worst case of the uncertain semi-infinite multiobjective optimization problem. We prove a necessary optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}} \), and then give a sufficient optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}}\). We formulate a Wolfe type dual problem of \({\mathrm{(RSIMP)}}\) and give duality results which hold between \({\mathrm{(RSIMP)}}\) and its dual problem.