aMUSE plans to strengthen and extend the collaboration between EU and US researchers to carry out cutting-edge searches for New Physics (NP) in the muon sector, while promoting the development of next generation muon accelerators. The project finds its roots in the previous MUSE network based at the Muon Campus of FNAL, USA. Here, the Muon (g-2) experiment aims to solve the long-standing muon anomaly puzzle and related NP implications collecting a twenty-fold increase in statistics compared to its predecessor. A high-profile discovery path for the search of charged Lepton Flavour Violation (cLFV) will be exploited by the Mu2e experiment, whose goal is to improve the discover sensitivity for the as yet unseen muon-to-electron conversion by four orders of magnitude, reaching mass scales of 10^4 TeV/c^2. aMUSE promotes an ambitious extension of the Muon Campus activities. Its R&D programs will exploit the future phase of Mu2e-II with an upgraded proton beam providing a ten-fold increase in muon yield. aMUSE aims to design and develop state-of-the-art detectors to face this challenge. At this high intensity frontier, aMUSE will explore the design of a beam line extension to seed the birth of new generation experiments searching for cLFV muon decays as a possible alternate running to Mu2e-II. Its final goal is to vastly improve sensitivity with respect to currently existing or proposed facilities and promote the integration of EU groups in developing accelerator and detector strategies. aMUSE further provides an excellent platform for an ambitious EU-US network to advance the development of muon beams. Low and high energy research is synergistic: muon cooling is fundamental to both high-energy muon collisions and low-energy high-intensity muon beams. The US longstanding competences in the study of muon beam technologies will be integrated with the experience of EU researchers, creating a unique opportunity for the advancement of a challenging and promising project.