The platform of a stairlift is moved along a rail in a stairwell. During the movement, the platform is automatically rotated relative to the rail about a vertical shaft, at angles depending on a position of the platform along the rail. The stairway contains, for instance, a virtually straight part and a bend, wherein the platform is rotated, at positions in the bend, at an orientation or orientations which make a smaller angle with a part of the rail going downstairs than an orientation of the platform in the straight part. In a stairwell with a wider part and a narrower part, wherein the stairwell is insufficiently wide to let the platform rotate through, at a position preceding the entering of the narrower part, the platform is rotated at an angle from which the platform can be rotated to a position for getting on and off in the narrower part without obstruction from walls of the stairwell.