Eventual presence of diamond carbon allotrope in space is discussed in numerous theoretical and experimental studies. The review summarises principal mechanisms of nanodiamond formation and experimental results of spectroscopic and structural investigations of nano- and microdiamonds from meteorites. Size dependence of diamond spectroscopic properties is discussed. Infrared spectroscopy allows detection of C-H bonds on surfaces of hot nanodiamond grains. Spectroscopic observation of nitrogen-related point defects in nanodiamonds is very challenging; moreover, such defects were never observed in nanodiamonds from meteorites. At the same time, photoluminescence and, eventually, absorption, of some impurity-related defects, in particular, of silicon-vacancy (SiV) center, observed in real meteoritic nanodiamonds, opens a possibility of diamond detection in astronomical observations.