The NNPDF Collaboration presents an extension of the existing NNPDF4.0 parton distribution function (PDF) sets to approximate next-to-next-to-next-to-leading order (aN³LO). They construct an approximation to the N³LO splitting functions using available partial information from fixed-order computations and small/large x resummations. The uncertainty in this approximation is estimated by varying the basis functions used. They include known N³LO corrections to deep-inelastic scattering structure functions and extend the FONLL general-mass scheme to O(αs³) accuracy. A set of aN³LO PDFs is determined by accounting for uncertainties in splitting functions and missing higher-order corrections (MHOU) through a theory covariance matrix formalism. The perturbative stability of the resulting PDFs is assessed, and their impact on Higgs and Drell-Yan cross-sections is examined. The aN³LO NNPDF4.0 PDFs are found to be consistent with NNLO counterparts within uncertainties, improving the description of global data and perturbative convergence of cross-sections. MHOU on PDFs decrease with increasing perturbative order. The paper discusses the construction of an approximate N³LO evolution, including the anomalous dimension matrix and its uncertainty, and the results of N³LO partonic cross-sections. It presents the main results of the work, including the aN³LO NNPDF4.0 PDF set, and compares them to the MSHT20 group. The paper also examines the phenomenological impact of aN³LO corrections on parton luminosities at the LHC and provides an assessment of the impact of aN³LO PDFs on Higgs and Drell-Yan cross-sections. The paper concludes with a summary and outlook on future developments. The results show good perturbative convergence across all x values, with MHOU decreasing with increasing accuracy. The aN³LO results agree well with small-x resummation results, indicating that the inclusion of aN³LO terms improves the quality of fixed-order NNLO fits. The paper also highlights the importance of small-x resummation for accurate predictions at very small x values.The NNPDF Collaboration presents an extension of the existing NNPDF4.0 parton distribution function (PDF) sets to approximate next-to-next-to-next-to-leading order (aN³LO). They construct an approximation to the N³LO splitting functions using available partial information from fixed-order computations and small/large x resummations. The uncertainty in this approximation is estimated by varying the basis functions used. They include known N³LO corrections to deep-inelastic scattering structure functions and extend the FONLL general-mass scheme to O(αs³) accuracy. A set of aN³LO PDFs is determined by accounting for uncertainties in splitting functions and missing higher-order corrections (MHOU) through a theory covariance matrix formalism. The perturbative stability of the resulting PDFs is assessed, and their impact on Higgs and Drell-Yan cross-sections is examined. The aN³LO NNPDF4.0 PDFs are found to be consistent with NNLO counterparts within uncertainties, improving the description of global data and perturbative convergence of cross-sections. MHOU on PDFs decrease with increasing perturbative order. The paper discusses the construction of an approximate N³LO evolution, including the anomalous dimension matrix and its uncertainty, and the results of N³LO partonic cross-sections. It presents the main results of the work, including the aN³LO NNPDF4.0 PDF set, and compares them to the MSHT20 group. The paper also examines the phenomenological impact of aN³LO corrections on parton luminosities at the LHC and provides an assessment of the impact of aN³LO PDFs on Higgs and Drell-Yan cross-sections. The paper concludes with a summary and outlook on future developments. The results show good perturbative convergence across all x values, with MHOU decreasing with increasing accuracy. The aN³LO results agree well with small-x resummation results, indicating that the inclusion of aN³LO terms improves the quality of fixed-order NNLO fits. The paper also highlights the importance of small-x resummation for accurate predictions at very small x values.