Provided is a fuel cell system which is capable of well preventing deterioration of performance of a fuel cell stack or well restoring deteriorated performance of the fuel cell stack and of suppressing deterioration of durability of the fuel cell stack, and a method of operating the fuel cell system. In a fuel cell system of the present invention including a fuel cell stack 100 in which an anode gas passage, a cathode gas passage, and a heat transmission medium passage have a structure in which inlet-side regions of the anode gas passage, the cathode gas passage, and the heat transmission medium passage substantially overlap with each other and outlet-side regions of the anode gas passage, the cathode gas passage, and the heat transmission medium passage substantially overlap with each other as viewed from a direction in which the unit cells are stacked, comprises at least one of an anode gas flow inverting device 122 for inverting a flow direction of the anode gas in the anode gas passage from a forward direction A to a reverse direction B which is opposite to the forward direction, and a cathode gas flow inverting device 123 for inverting a flow direction of the cathode gas in the cathode gas passage from the forward direction A to the reverse direction B; a heat transmission medium flow inverting device 124 for inverting a flow direction of the heat transmission medium in the heat transmission medium passage from the forward direction A to the reverse direction B; and a control system 300 configured to control the inverting devices 122, 123, and 124.