The work described in this paper concerns a novel method for directly forming curved profiles/sections from billets in one extrusion operation using two opposing punches. Its mechanics are based on internal differential material flow, and it has been given the acronym, differential velocity sideways extrusion (DVSE). A tool set enabling sideways extrusion to be performed using opposing punches moving with different velocities was used for a series of experiments in which punch velocity ratio and extrusion ratio were process parameters. Plasticine was used as a model work-piece material and a series of compression tests were undertaken, to determine its constitutive properties and gain an estimate of work-piece die friction for use in process simulation. Curvature of extrudate can be controlled and varied using a difference between the velocities of the two punches, defined by velocity ratio. Greater curvature is achieved with lower velocity ratio. Curvature is also dependent on extrusion ratio, an increase in which increases curvature, although curvature is less sensitive to it than to velocity ratio. The extent of work-piece flow velocity gradient across the die exit orifice, which causes curvature, has been identified. Severe plastic deformation of the extrudate occurs in a way similar to channel angular extrusion (CAE), thus a greatly promoted effective strain level is achieved, though it is not always uniform across a section. The inner bending region of an extrudate experiences maximum localised effective strain, which decreases with decrease in curvature. To the authors' knowledge this is the first publication in which extrudate curvature is deliberately induced using opposing punches with differential velocities. Although only fixed velocity ratio values have been used in the work described in this paper the ability to change during operation exists and the process has the potential for the production of a profile with different curvature along its length.