Topological states of quantum matter have attracted great attention in condensed matter physics and materials science. The study of time-reversal-invariant (TRI) topological states in quantum materials has made tremendous progress in both theories and experiments. As a great success, thousands of TRI topological materials are predicted through sweeping search. Richer exotic phenomena are expected to appear in magnetic topological materials because of varied magnetic configurations, but this study falls much behind due to the complex magnetic structures and transitions. Here, we predict the tetradymite-type compound MnBi$_2$Te$_4$ and its related materials host interesting magnetic topological states. The magnetic ground state of MnBi$_2$Te$_4$ is an antiferromagnetic phase which leads to an antiferromagetic topological insulator state with a large topologically non-trivial energy gap ($sim$0.2~eV). It is the parent state for the axion state, which has gapped bulk and surface states, and quantized topological magnetoelectric effect. The ferromagnetic phase of MnBi$_2$Te$_4$ leads to an ideal minimal type-II Weyl semimetal with two Weyl points accompanied by one hole-type and one electron-type Fermi pocket at the Fermi level, which has never been discovered elsewhere. We further present a simple and unified continuum model to capture the salient topological features of this kind of materials.
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