In the last years there has been a great interest in magnetic systems formed by a cluster of transition metal ions covalently bonded via superexchange bridges, embedded in a large organic molecule. Following the synthesis and the structural and magnetic characterization of these magnetic molecules by chemists, the physicists realized the great interest of these systems as a practical realization of zero-dimensional model magnetic systems. In fact the magnetic molecules can be synthesized in crystalline form whereby each molecule is magnetically independent since the intramolecular exchange interaction among the transition metal ions is dominant over the weak intermolecular, usually dipolar, magnetic interaction. We have undertaken a systematic NMR investigation of molecular nanomagnets since back in 1996. The present review tries to give an account of the main results obtained so far and of the many exciting projects that still lie ahead. The work was done through a continuous very fruitful collaboration among three NMR laboratories: at the University of Pavia, Italy, at Iowa State University and Ames Laboratory, Ames, IA, USA and at Hokkaido University, Sapporo, Japan with occasional very useful collaborations with the high field NMR lab. in Grenoble, France. None of the work could be done without the precious collaboration and help of our colleagues in chemistry at the University of Florence and of Modena, Italy and at Ames Laboratory in USA who synthesized and characterized the samples used in the NMR work.