The epitaxially grown magnetic nanostructures including nanodots, nanowires and nanorings have been attracting much scientific and engineering interests because of their expected unique physical characteristics due to quantum effects. These epitaxial nanomagnets and their array are undoubtedly thought to make major contribution to the development of future SPINTRONICS devices, ultra-high density magnetic random access memory (MRAM) and magnetic switching devices for examples, and other quantum devices. In this case, epitaxial growth of the nanomagnets and the resulting anisotropic properties are one of the largest interest as well as fine-nanostructuring. There have been some concerns such as throughput rate with conventional nanoprocessing techniques involving FIB lithography and e-beam lithography, and/or minimization-limit with photolithography due to the wavelength. On the other hand, self-assembly or self-organized methods could also be used for construction of nanopatterns, in which such nanostructures are directly built up from separate atoms. Here we report about formation and characterization of self-organized nanomagnet arrays made of metals and oxides. We have epitaxially grown ferrimagnetic Fe3O4 (111), (Mn0.55Zn0.35Fe0.10)Fe2O4 (111), ferromagnetic Ni (111) and antiferromagnetic NiO (111) nanodots, nanowires and nanogroove arrays on the atomically stepped ultra-smooth sapphire (0001) substrate by LaserMBE. The sapphire (a-Al2O3 single crystal) substrates have atomic steps of 0.2 nm in height and atomically flat terraces of 50-100 nm in width so that self-assembly processes of nanomagnet arrays were strongly induced by the energetic instability at the straight and periodic step-edges. In this study, NiO worked as a antiferromagnetic exchange bias layer. On the other hand, NiO was also reduced into ferromagnetic Ni metal by annealing in hydrogen atmosphere in some situations for further applications. Crystallographic and morphology analyses of the nanomagnets were made by in-situ reflection high-energy electron diffraction (RHEED), ex-situ X-ray diffraction (XRD), transmission electron microscope (TEM). and atomic force microscopy (AFM). Magnetic properties were characterized by superconducting quantum interference device (SQUID) magnetometer and magnetic force microscopy (MFM). Further experimentals are conducted for magneto-optical characterizations for above mentioned metal and/or oxide nanomagnet arrays.