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Low-level concentrations of primordial and cosmogenic radionuclides were determined in recently fallen chondrites and in meteorites from Mars and the Moon. The radionuclide measurements showed a wide range of concentrations of $^{22}Na$, $^{26}Al$, $^{46}Sc$, $^{54}Mn$ and $^{57}Co$. The absence of the neutron product ($^{60}Co$) in some of the meteorites indicates that their pre-atmospheric radius was small (on the order of 10 cm). This has also been confirmed by Monte Carlo simulations of the production rates of investigated cosmogenic radionuclides, and their comparison with experimental data. The cosmic-ray exposure ages of the investigated meteorites have been estimated to range from 1 to 5 million years. The agreement between the simulated and observed $^{26}Al$ activities indicates that the meteorites were mostly irradiated by a long-term average flux of galactic cosmic rays.