Analyses of Lunar Laser ranges show a displacement in direction of the Moon's pole of rotation which indicates that strong dissipation is acting on the rotation. Two possible sources of dissipation are monthly solid-body tides raised by the Earth (and Sun) and a fluid core with a rotation distinct from the solid body. Both effects have been introduced into a numerical integration of the lunar rotation. Theoretical consequences of tides and core on rotation and orbit are also calculated analytically. These computations indicate that the tide and core dissipation signatures are separable. They also allow unrestricted laws for tidal specific dissipation Q versus frequency to be applied. Fits of Lunar Laser ranges detect three small dissipation terms in addition to the dominant pole-displacement term. Tidal dissipation alone does not give a good match to all four amplitudes. Dissipation from tides plus fluid core accounts for them. The best match indicates a tidal Q which increases slowly with period plus a small fluid core. The core size depends on imperfectly known properties of the fluid and core-mantle interface. The radius of a core could be as much as 352 km if iron and 374 km for the Fe-FeS eutectic composition. If tidal Q versus frequency is assumed to be represented by a power law, then the exponent is -0.19¿0.13. The monthly tidal Q is 37 (-4, +6), and the annual Q is 60 (-15, +30). The power presently dissipated by solid body and core is small, but it may have been dramatic for the early Moon. The outwardly evolving Moon passed through a change of spin state which caused a burst of dissipated power in the mantle and at the core-mantle boundary. The energy deposited at the boundary plausibly drove convection in the core and temporarily powered a dynamo. The remanent magnetism in lunar rocks may result from these events, and the peak field may mark the passage of the Moon through the spin transition. ¿ 2001 American Geophysical Union