High energy colliders offer the opportunity to widen our investigation of sub-nuclear phenomena to the highest unexplored energy regimes and shortest interaction scales. They consequently have the strongest potential for discovering new heavy particles and forces, allowing more accurate precision tests of Standard Model objects, and detecting new elusive signatures like the ones associated with dark matter particles.
In 2014-2015 the INFN National Scientific Committee for High Energy Physics with Accelerator (CSN1) discussed and prepared an extensive document  to describe the present understanding of the scientific scenarios and to propose recommendations towards future experiments at high energy particle accelerators.
Dedicated physics studies and R&D on innovative technologies are longstanding activities for the CSN1 community of over one thousand physicists and engineers. This valuable work has been exploited as a contribution in the context of international collaborations, preparing the dedicated inputs for the update of the European Strategy for Particle Physics.
The present document is the outcome of several workshops and discussions, during the past three years, devoted to study different opportunities both at the high energy and high intensity frontiers.