A laser processing apparatus comprising: - at least one first laser device, each first laser device connected to at least one first optical feed fiber for delivering a first laser beam; - at least one second laser device, each second laser device connected to at least one second optical feed fiber for delivering a second laser beam; and characterised by the following: - beam combining means connected to said first and second feed fibers and to a multi-core optical fiber (35; 50), the combining means being adapted to form a composite laser beam by having said at least one first optical feed fiber (72; 56) aligned with a first core of said multi-core optical fiber and said at least one second optical feed fiber (71; 57) aligned with at least one second core of said multi-core optical fiber, wherein said first core of said multi-core optical fiber has a circular cross-section, said second core has an annular shape concentric to said first core, and the beam combining means comprise a multi-bore capillary tube having a body portion comprising a glass tube, an input end for receiving laser beams carried by said optical feed fibers, and an opposite output end for delivering the composite output laser beam, wherein said optical feed fibers, entering at the input end extend through the body portion in capillary bores to the output end, are fused with the glass tube to form a component consisting of light guiding cores (71a, 72a) and encircling glass material; - said first and second cores being connectable to a laser processing head, to direct a composite laser beam comprising a first output laser beam and a second output laser beam from the laser processing head to a workpiece to be cut, wherein the first output laser beam has a circular cross section and the second output laser beam has annular cross-section spaced from and concentric to the first laser output beam at the workpiece; and - a control unit functionally connected to said first and second laser devices (30, 31), to individually control the power density in said first and second output laser beams (1, 2), wherein the control unit is configured to adapt the power density of at least one of the first and second laser beams (1, 2) in response to approaching a change point in direction of cutting progression, and to adapt the power density in accordance with the speed of the laser processing head relative to the workpiece. - at least one first laser device, each first laser device connected to at least one first optical feed fiber for delivering a first laser beam; - at least one second laser device, each second laser device connected to at least one second optical feed fiber for delivering a second laser beam; - beam combining means connected to said first and second feed fibers and to a multi-core optical fiber (35; 50), the combining means being adapted to form a composite laser beam by having said at least one first optical feed fiber (72; 56) aligned with a first core of said multi-core optical fiber and said at least one second optical feed fiber (71; 57) aligned with at least one second core of said multi-core optical fiber, wherein said first core of said multi-core optical fiber has a circular cross-section, said second core has an annular shape concentric to said first core, and the beam combining means comprise a multi-bore capillary tube having a body portion comprising a glass tube, an input end for receiving laser beams carried by said optical feed fibers, and an opposite output end for delivering the composite output laser beam, wherein said optical feed fibers, entering at the input end extend through the body portion in capillary bores to the output end, are fused with the glass tube to form a component consisting of light guiding cores (71a, 72a) and encircling glass material; - said first and second cores being connectable to a laser processing head, to direct a composite laser beam comprising a first output laser beam and a second output laser beam from the laser processing head to a workpiece to be cut, wherein the first output laser beam has a circular cross section and the second output laser beam has annular cross-section spaced from and concentric to the first laser output beam at the workpiece; and - a control unit functionally connected to said first and second laser devices (30, 31), to individually control the power density in said first and second output laser beams (1, 2), wherein the control unit is configured to adapt the power density of at least one of the first and second laser beams (1, 2) in response to approaching a change point in direction of cutting progression, and to adapt the power density in accordance with the speed of the laser processing head relative to the workpiece.