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Combining robust magnetism, strong spin-orbit coupling and uniquethickness-dependent properties of van der Waals crystals could enable newspintronics applications. Here, using density functional theory, we propose the(MnSb$_2$Te$_4$)$\cdot$(Sb$_2$Te$_3$)$_n$ family of stoichiometric van derWaals compounds that harbour multiple topologically-nontrivial magnetic phases.In the groundstate, the first three members of the family, i.e. MnSb$_2$Te$_4$,($n=0$), MnSb$_4$Te$_7$, ($n=1$), and MnSb$_6$Te$_{10}$, ($n=2$), are 3Dantiferromagnetic topological insulators (AFMTIs), while for $n \geq 3$ aspecial phase is formed, in which a nontrivial topological order coexists witha partial magnetic disorder in the system of the decoupled 2D ferromagnets,whose magnetizations point randomly along the third direction. Furthermore, dueto a weak interlayer exchange coupling, these materials can be field-driveninto the FM Weyl semimetal ($n=0$) or FM axion insulator states ($n \geq 1$).Finally, in two dimensions we reveal these systems to show intrinsic quantumanomalous Hall and AFM axion insulator states, as well as quantum Hall state,achieved under external magnetic field, but without Landau levels. Our resultsprovide a solid computational proof that MnSb$_2$Te$_4$, is not topologicallytrivial as was previously believed that opens possibilities of realization of awealth of topologically-nontrivial states in the(MnSb$_2$Te$_4$)$\cdot$(Sb$_2$Te$_3$)$_n$ family.

Topological magnetic materials of the (MnSb$_2$Te$_4$)$\cdot$(Sb$_2$Te$_3$)$_n$ van der Waals compounds family