The first rocks to be returned from the Moon by the Apollo 11 astronauts were basalts from the mare basins. Analysis of these rocks led to the hypothesis that the mare lavas were remelts of a lunar interior that had experienced an early, profound chemical differentiation event produced by crystallization of a planet-wide lunar magma ocean. As Apollo missions continued to explore and sample the lunar surface, an increasingly diverse suite of mare volcanic rocks was discovered. Mare magmatism is concentrated in the time interval of 3.8 to 3.0 billion years before present. Among the samples were tiny, glassy spheres of ultramafic composition that formed during volcanic fire-fountain eruptions into the cold lunar vacuum. The results of high-pressure and high-temperature laboratory melting experiments on lunar mare basalts and volcanic glasses, along with geochemical evidence and physical modeling, affirm that remelting of the solidified products of a deep magma ocean still provides the best explanation for lunar maria magmas. However, the initial depth of the lunar magma ocean, the physical processes that accompanied solidification, and the heat source for remelting cumulates to form these late basaltic outpourings remain incompletely understood and present challenging problems for current researchers.