The discovery and characterization of planets of young stars represents an important goal as it will allow to put much stronger observational constraints on the current theories of planet formation and orbital evolution. This thesis reports the results of an optical spectroscopic survey of a sample of 43 young (30-300 Myr), late spectral type, nearby stars. Observations have been carried out with the 2m Schmidt telescope and the Coudé spectrograph at the TLS-Tautenburg Observatory, and were aimed to measure high precision radial velocities (RV). In spite of their enhanced level of photospheric activity inducing RV-jitters, the survey demonstrates that also young stars are suitable targets for the detection of planets by means of the RV-technique. In fact most of the stars in the sample show a RV-scatter < 30 m/s Seven spectroscopic binaries have been detected, and for 2 of them the best-fit RV-curve has been calculated and the orbital parameters derived. No planet has been found with a sufficiently high level of confidence. There is only one target showing a RV-signal compatible with a planet of mass M = 0.16 Mj (Jupiter mass), which deserves further investigations. The large number of RV measurements (between 30 and 50 for most of the stars) allows to put stringent upper limits to the presence of planets around the surveyed stars. For instance, a planet with period P < 10 days and minimum mass Msin(i) > 2 Mj, can be excluded for 80 % of the stars (95 % if we consider Msin(i) > 7 Mj).