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I present recent progress in global PDF analyses within the NNPDF framework, focusing on fits of PDFs and the strong coupling at approximate N3LO (aN3LO) accuracy, and their implications for LHC phenomenology. I also discuss how to combine the MSHT20 and NNPDF4.0 aN3LO fits and assess the consequences for Higgs production cross-sections at the LHC Run-3 and beyond. I also briefly discuss PDFs tailored for Monte Carlo event generators at the LHC