A razor blade unit with a blade housing, in which several blades with cutting edges lying parallel to a skin contact plane are arranged, a coupling section, which can be connected preferably in a detachable manner to a handle, and a connecting section. The connecting section includes at least one film hinge between the blade housing and the coupling section, which hinge defines a pivot axis extending parallel to the skin contact plane, and an elastic damping component, which surrounds one or more sides of the film hinge and is formed both on the blade housing and on the coupling section. The razor blade unit also has a foam edge, arranged in front of the cutting edges (with respect to the shaving direction) of the blades to pretension the skin during shaving. The pivot axis extending through the film hinge is positioned between the forward boundary (with respect to the shaving direction) of the foam edge and the cutting edge lying closest to the foam edge.