At astrophysically relevant temperatures, nuclear cross-sections are extremely small and experimental measurements in a laboratory at the Earth's surface are hampered by the cosmic background. The LUNA Collaboration has exploited the unique features of the rock cover offered by the LNGS underground laboratory in terms of background reduction, to study very important H-burning reactions at astrophysically relevant energies. After a general introduction on the LUNA experiment, the most important results and their astrophysical consequences will be reviewed. On-going and future measurements will also be presented.