Interaction in a flat band is magnified due to the divergence in the density of states, which gives rise to a variety of many-body phenomena such as ferromagnetism and Wigner crystallization. Until now, however, most studies of the flat band physics are based on model systems, making their experimental realization a distant future. Here, we propose a class of systems made of real atoms, namely, carbon atoms with realistic physical interactions (dubbed here as Kagome graphene/graphyne). Density functional theory calculations reveal that these Kagome lattices offer a controllable way to realize robust flat bands sufficiently close to the Fermi level. Upon hole doping, they split into spin-polarized bands at different energies to result in a flat-band ferromagnetism. At a half filling, this splitting reaches its highest level of 768 meV. At smaller fillings, e.g., when = 1 6 , on the other hand, a Wigner crystal spontaneously forms, where the electrons form closed loops localized on the grid points of a regular triangular lattice. It breaks the translational symmetry of the original Kagome lattice. We further show that the Kagome lattices exhibit good mechanical stabilities, based on which a possible route for experimental realization of the Kagome graphene is also proposed.Recently, fractional Chern insulators based on the flat band structure have also been extensively studied in lattice systems in the absence of magnetic Landau levels [28][29][30].Kagome lattice is a lattice which can generate flat band. To date, however, most studies on its flat band physics are based on toy models [31][32][33]. Experimentally, available materials with a Kagome lattice are mostly frustrated magnets [34][35][36][37][38][39], which are half-filled Mott insulators.Unfortunately, Fermi level ( ) in such materials is distance away from the flat bands. Given the rich bonding chemistry but simple band structure of carbon, e.g., for graphene only pz states exist near , and exceptional stabilities of its various allotropes, one may ask if it is possible to synthesize carbon-based Kagome or Kagome-like lattices and what would be the physics of strong correlation in them.In this paper, we identify, by first-principles total-energy calculations, a family of twodimensional (2D) carbon Kagome-like structures (to be termed Kagome graphene/graphyne), which can host flat bands and relevant physical phenomena. The most basic structure is the Kagome graphene made of carbon triangles on a regular honeycomb lattice. Noticeably, flat bands appear near in all these structures. Upon hole doping, the spin degeneracy of the flat bands is spontaneously lifted to result in a flat-band ferromagnetism with a large spin splitting. At a 1/6 filling factor of the flat bands, a Wigner crystallization is found, while at other filling factors, various charge density wave patterns are expected. These interesting physical phenomena are interpreted based on a Hubbard model. The possibility of realizing an anomalous quantum Hall effect, as well as the possible routes t...