This paper introduces the concept of cross-generational system, which indicates an information technology system designed to allow different generations interacting through centuries or, even, millennia. In the era of digital immortality and advances in artificial intelligence, there will be not only the problem of preserving and accessing data, but the emergence of digital clones will bring new challenges for technology development, interaction design and ethics. This paper provides an overview of societal transformation towards digital immortality, then describes the vision of cross-generational system design, investigates the relative future challenges and proposes the eventual socio-ethical questions.