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The stellar evolution and chemical make-up of the Universe are determined by nuclear reactions occurring in a wide variety of stellar sites. Precise determinations of the cross sections of these reactions are crucial for the calculation of reaction rates and for the development of stellar evolution models. The Laboratory for Underground Nuclear Astrophysics (LUNA) collaboration has been at the forefront of the direct measurement of nuclear reactions at the low energies of astrophysical interest for the last 35 years. The many significant results achieved at LUNA have been made possible due to the low background conditions uniquely available thanks to its location deep underground at the Laboratori Nazionali del Gran Sasso. Another key aspect of these successes is due to the experience of the LUNA collaboration in the production and characterization of a variety of solid targets used in reaction measurements. In this review, the main production techniques of solid targets are described, as well as the common methods adopted for target degradation monitoring. We also present the results of recent measurements using these targets and the future plans of the LUNA collaboration for measurements using solid targets at the LUNA400 kV and the new Ion Beam Facility (IBF) 3.5 MV are also presented.