The invention relates to a process for making a bio-diesel and a bio-naphtha from a complex mixture of natural occurring fats & oils, wherein said complex mixture is optionally subjected to a refining treatment for removing the major part of the non-triglyceride and non-fatty acid components, thereby obtaining refined oils; said complex mixture or refined oils are subjected to a hydrolysis step for obtaining glycerol and a mixture of free fatty acids; said mixture of free fatty acids are subjected to a fractionation step for obtaining: a liquid or substantially liquid free fatty acids part (phase L); and a solid or substantially solid free fatty acids part (phase S); and said phase L is transformed into alkyl-esters as bio-diesel by an esterification; said phase S is transformed into linear or substantially linear paraffins as the bio-naphtha: by hydrodeoxygenation or decarboxylation of the free fatty acids; or from said phase S are obtained fatty acids soaps that are transformed into linear or substantially linear paraffins as the bio-naphtha by decarboxylation of the soaps.