An elevator with a brake device for braking a car and/or keeping it stationary on at least one braking rail. The brake device has at least two brake shoes situated opposite from each other, and at least one brake clamp with two clamp arms for opening and closing clamp jaws acting on the brake shoes. In certain embodiments, only one of the clamp arms is connected to the car in pivoting fashion via a bearing with a pivot axle that is affixed to the car, oriented parallel to the direction of travel. In certain embodiments, the brake clamp is supported on the car in such a way that the clamp jaws, by pivoting together in the same direction, are able to follow a movement that is forced on a brake shoe due to local position deviations of the associated braking rail.