In this position paper, we discuss two relevant topics: (i) generic multiscale computing on emerging exascale high-performing computing environments, and (ii) the scaling of such applications towards the exascale. We will introduce the different phases when developing a multiscale model and simulating it on available computing infrastructure, and argue that we could rely on it both on the conceptual modelling level and also when actually executing the multiscale simulation, and maybe should further develop generic frameworks and software tools to facilitate multiscale computing. Next, we focus on simulating multiscale models on high-end computing resources in the face of emerging exascale performance levels. We will argue that although applications could scale to exascale performance relying on weak scaling and maybe even on strong scaling, there are also clear arguments that such scaling may no longer apply for many applications on these emerging exascale machines and that we need to resort to what we would call multi-scaling . This article is part of the theme issue ‘Multiscale modelling, simulation and computing: from the desktop to the exascale’.