The piezoelectric damping device is a passive piezoelectric vibration suppression and damping device for reducing amplitude of vibration in structures. The piezoelectric damping device includes a magnetic layer having opposed inner and outer faces, with the inner face thereof being adapted for releasable magnetic attachment to a vibrating structure, such as a metallic pipe, for example. A piezoelectric layer is secured to the outer face of the magnetic layer. Preferably, an electrically insulating layer is sandwiched between the magnetic layer and the piezoelectric layer. The electrically insulating layer is preferably an electrically insulating adhesive. An electrical shunting circuit is in electrical communication with the piezoelectric layer, such that vibration in the piezoelectric layer caused by the vibrating structure generates electrical energy, which is then dissipated by the electrical shunting circuit, thus damping vibration in the vibrating structure.