NOVELTY - The system has a fuel cell (2) i.e. solid oxide fuel cell (SOFC), comprising an anode (3) and a cathode (4) between which an electrolyte (5) is arranged. An afterburner (25) is provided for burning fuel, which is contained in anode exhaust gas, using an oxygen-containing gaseous mixture i.e. air. An anode exhaust system (26) conducts the anode exhaust gas to the afterburner, and a gas conveyer unit (17) e.g. blower, supplies the oxygen-containing gaseous mixture to the afterburner and optionally the fuel cell and stands in direct gas-conductive contact with the afterburner.    USE - Fuel cell system e.g. stationary fuel cell system such as SOFC system, and mobile fuel cell system, for use in a motor vehicle and a heating system.    ADVANTAGE - The gas conveyer unit supplies the oxygen-containing gaseous mixture to the afterburner and optionally the fuel cell and stands in direct gas-conductive contact with the afterburner, thus enabling optimal, safe, efficient, adjustable and flexible combustion of the anode exhaust gas and/or its components in an energy-saving manner, achieving high electrical efficiency and durability of the fuel cells. The system achieves increase in pressure caused by the afterburner does not have an effect on pressure level within the fuel cell, so that operation of the fuel cell is not interfered by adaption and/or optimization of conditions in the afterburner, enhances behavior of the afterburner in partial load range and optimizes incineration process regarding emissions, thus reducing wear of the system, hence extending maintenance intervals of the system, and enabling operation of the system to be performed in an inexpensive manner.    DETAILED DESCRIPTION - An INDEPENDENT CLAIM is also included for a method for operating a fuel cell system.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic connection diagram of components of a fuel cell system.    Fuel cell (2)    Anode (3)    Cathode (4)    Electrolyte (5)    Gas conveyer unit (17)    Afterburner (25)    Anode exhaust system (26)