The invention relates to a burner (13) with surface combustion, having a first and at least one additional burner zone (34, 35), each of which has a feed (29, 37) for a fuel/air mixture in a combustion zone (51) and each of which has a woven membrane (49) for each burner zone (34, 35), the combustion zone (51) adjoining said fabric membrane. The burner also comprises a flashback barrier (48) which is arranged upstream of the woven membrane (49) and at a distance to the fabric membrane. Each burner zone (34, 35) has a separate feed (29, 37) for the fuel/air mixture, and each burner zone (34, 35) has a flame matrix (52), which are aligned in the same direction, in the combustion zones (51).