Artificial intelligence (AI) is gaining momentum in the scientific and industrial community for the ever-growing number of applications where such innovative techniques of learning form and processing large amount of data have proved successful. High-performance computing (HPC) and cloud resources providers are moving faster to be able to support new applications that benefit from the combination of traditional HPC simulation, machine learning and deep learning processing and big data analytics. However, the tighter the combination of these three elements is, the more complex the integration of innovative architectures into a single execution platform becomes. On one hand, application workflow management systems need to incorporate more functionalities and support dynamism in the execution, by preserving (energy) efficiency of the infrastructural resources. On the other hand, more exotic hardware accelerators (ranging from GPUs and FPGAs, to neural network processors (NNPs), to neuromorphic processors) need to be integrated in the computing assets in order to leverage performance boost. This chapter provides an overview of the future HPC, AI, and big-data cross-stack execution platform, as devised in the funded EuroHPC ACROSS project, which will be tailored to cope with all these challenges, and to support future exascale-ready applications.