The Alpha Magnetic Spectrometer is a general-purpose particle physics detector operating on the International Space Station. Precision measurements by AMS of the cosmic-ray elementary particle fluxes and nuclei fluxes reveal new unexpected phenomena. The positron flux exhibits a significant excess starting from $25.2 \pm 1.8$ GeV followed by a sharp drop-off above $284^{+91}_{-64}$ GeV, consistent with a primary source of cosmic-ray positrons from either dark matter collisions or new astrophysical sources. The different behavior of the cosmic-ray electron flux and positron flux shows that most high energy electrons originate from different sources than high energy positrons. Intriguingly, the positron flux and the antiproton flux have strikingly similar behavior at high energies. New observations from AMS on cosmic nuclei show that primary cosmic-ray He, C, and O have an identical rigidity dependence above 60 GV and deviate from a single power-law above 200 GV. Unexpectedly, the primary Ne, Mg, and Si also have an identical rigidity dependence above 86.5 GV, but they are different from that of He, C, and O. This shows that primary cosmic rays have at least two distinct classes of rigidity dependence. Above 30 GV, secondary cosmic nuclei Li, Be, and B have identical rigidity dependence which is distinctly different from those of primary cosmic rays. The results from AMS on many different types of cosmic rays are not explained by the current theoretical models and provide unique input to the understanding of the origins and evolution of cosmic rays in the galaxy.