Topological states of quantum matter have attracted great attention in condensed matter physics and materials science. The study of time-reversal-invariant topological states in quantum materials has made tremendous progress. However, the study of magnetic topological states falls much behind due to the complex magnetic structures. Here, we predict the tetradymite-type compound MnBi_{2}Te_{4} and its related materials host topologically nontrivial magnetic states. The magnetic ground state of MnBi_{2}Te_{4} is an antiferromagetic topological insulator state with a large topologically nontrivial energy gap (∼0.2 eV). It presents the axion state, which has gapped bulk and surface states, and the quantized topological magnetoelectric effect. The ferromagnetic phase of MnBi_{2}Te_{4} might lead to a minimal ideal Weyl semimetal.