Lignocellulosic biomass can be fractionated for the purpose of increasing cellulose purity in the pulp, increasing native lignin content of the isolated lignin, and improving cellulose hydrolysis, by performing the steps of: (a) extracting the biomass with an extracting liquid comprising at least 20 wt % of a first organic solvent at a temperature below 100° C.; (b) treating the extracted biomass with a treatment liquid comprising a second organic solvent selected from lower alcohols, ethers and ketones, optionally water and optionally an acid, at a temperature between 120° C. and 280° C., and, optionally: (c) subjecting a cellulose-enriched product stream resulting from step (b) to enzymatic hydrolysis. The first and second organic solvent may be different or the same; in particular they comprise ethanol or acetone.