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 flat-band physics have been based on model systems, making their experimental realization something that would occur in the distant future. Here, we propose a class of systems made of real atoms, namely carbon atoms with realistic physical interactions (dubbed here "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 half-filling, this splitting reaches its highest level of 768 meV. At smaller fillings, e.g., when nu = 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.