1. A pendulum mass damper for damping oscillations of a structure, the pendulum mass damper comprising a mass (1) balanced by a first spring system (2a, 2b, 2c) that balances the mass (1) horizontally and is positioned above or at substantially a same vertical level as the mass (1), wherein the mass is supported by a carrying part (4) to maintain a vertical position, the carrying part (4) carrying the mass (1) in the vertical direction extends between the mass (1) and a position (C) below the mass (1), and therefore the weight of the mass (1) is carried or supported from or at a point or level below the mass (1), such that the mass (1) at the position (C) below the mass is connected to a unit (5) constituting a base of a supporting system for the mass (1), the unit (5) balanced by flexible parts (8) such that the unit (5) can move either horizontally or both horizontally and vertically.