Systems and methods for producing mesh from wires or rods with programmed changeable steps for the longitudinal and transverse wires. The longitudinal wires (1) and the transverse wires (12) may be fed from coils or be precut. The longitudinal wires are fed in receptacles (2) on carriers (3) with the carriers being found on prefeeder carrier (4), a feeder carrier (6) with grippers (7) transports them towards the welding heads (10) and the produced mesh (20) is received by a mesh carrier (14). The carriers (3) with the receptacles (2) for the longitudinal wires on the prefeeder carrier (4), the grippers for the longitudinal wires (7) at the feeder carrier (6) and the welding heads (10) are displaced in the direction of the transverse wire without restrictions, generally in an unrestricted fashion, so as to correspond to the longitudinal wires being subjected to welding. The transverse wires are fed towards the welding heads to be welded with the longitudinal wires. The machine produces meshes with openings, grouping the longitudinal wires in groups and feeding the groups of longitudinal wires towards the welding heads, adjusting the position of the related mechanisms to the position of the longitudinal wires.