The present invention provides a damping device with which a vibration damping action is improved and also provides a metal heat shield with which a sound insulation performance is improved. The damping device (21) includes: a substantially annular damping member (40) that has an insertion hole, through which the bolt (7) to be attached to an attachment boss (10) is inserted, and externally surrounds the bolt (7); a second retaining section (44) for retaining the damping member (40); and a coupling section (45) for coupling the first retaining section (43) and the second retaining section (44), the second retaining section (44), the coupling section (45), and the first retaining section (43) being provided in this order from the attachment boss (10) side; and a collar member (22) that includes a coupling section (27), which couples the pair of flange sections (28), (4) each other and has a gap in a radial direction corresponding to the damping member (40); the flange sections nip the damping member (40) from both sides in an axial direction with gaps formed on both the sides.