NNPDF progress and the path to proton structure at N3LO accuracy

We review recent progress in the global determination of the collinear parton distributions (PDFs) of the proton within the NNPDF framework. This progress includes NNPDF4.0 variants with QED effects and with missing higher order uncertainties (MHOUs) via the theory covariance matrix formalism, as well as PDFs based on QCD calculations at approximate N3LO (aN3LO) accuracy. We present the combination of these theoretical developments resulting into NNPDF4.0 aN3LO variants accounting for QED corrections and with a photon PDF. We compare these aN3LO QED PDFs with analogous results obtained by the MSHT group, and briefly quantify their implications at the level of representative LHC cross-sections.