Since 1991, LUNA (Laboratory for Underground Nuclear Astrophysics) has played a leading role in the direct measurement of nuclear cross sections relevant to astrophysics, producing several pioneering results. Thanks to the shielding provided by the Gran Sasso mountain, LUNA enables measuring very small cross sections of the nuclear reactions involved in stellar and Big Bang nucleosynthesis at (or very close to) the energy range at which they occur in astrophysical scenarios. The present accelerator is provided with two beamlines, respectively dedicated to a solid target and to a differential-pumping extended gas target. The most recent measurements are briefly presented, together with the experimental techniques used, and the astrophysical implications. The imminent installation of a new 3.5 MV accelerator paves the way for the study of a number of key reactions regarding helium and carbon burning.