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The article discusses the emerging non-von Neumann computer architectures and their integration in the computing continuum for supporting modern distributed applications, including artificial intelligence, big data, and scientific computing. It provides a detailed summary of the available and emerging non-von Neumann architectures, which range from power-efficient single-board accelerators to quantum and neuromorphic computers. Furthermore, it explores their potential benefits for revolutionizing data processing and analysis in various societal, science, and industry fields. The paper provides a detailed analysis of the most widely used class of distributed applications and discusses the difficulties in their execution over the computing continuum, including communication, interoperability, orchestration, and sustainability issues.