Lunar volcanism, as is generally accepted, started well before the emplacement of the mare fill at 3.1-3.9 Ga b.p. and likely extended well into the Eratosthenian (3.1-1.1 Ga b.p.). The early volcanism is relatively easily understood. The extension of the activity to, possibly, 1.5 Ga b.p., albeit at a decreasing rate, poses a problem since a relatively small body such as the Moon may be expected to cool rapidly and freeze a partially molten mantle layer quickly. We present thermal history models in which a region of partial melt forms in the mantle underneath the lithosphere almost immediately after the start of the model at depths between 300 and 700 km depending on the chosen initial depth of the magma ocean. To calculate the thermal histories we use axisymmetric 2D and fully 3D spherical shell convection codes with viscosity depending on the azimuthally averaged temperature. The thermal evolution of the Moon is found to be characterized by the growth of a massive 700to 800-km-thick lithosphere while the lower mantle and core cool only by 100-200 K. The partial melt zone decreases in thickness with time from top to bottom until it vanishes at times between 3.4 and 2.2 Ga b.p. The maximum degree of partial melting is between roughly 10 and 20%. The partial melt layer is initially global and is disrupted by cold downwellings as cooling proceeds. The melt zone freezes from above due to the thickening of the lithosphere, which implies that the source region of volcanic rock proceeds to increasing depth with time. This corresponds nicely with the variation of titanium with age of lunar mare basalts. Mixing of the melt zone with the underlying mantle chemically rejuvenates the melt zone to some degree and, thus, the source region of the mare basalts.