The recent demonstrations of electrical manipulation and detection of antiferromagnetic spins have opened up a new chapter in the story of spintronics. Here, we review the emerging research field that is exploring the links between antiferromagnetic spintronics and topological structures in real and momentum space. Active topics include proposals to realize Majorana fermions in antiferromagnetic topological superconductors, to control topological protection and Dirac points by manipulating antiferromagnetic order parameters, and to exploit the anomalous and topological Hall effects of zero-net-moment antiferromagnets. We explain the basic concepts behind these proposals, and discuss potential applications of topological antiferromagnetic spintronics.
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L.Š. acknowledges support from the Grant Agency of Charles University, no. 280815, and EU FET Open RIA Grant 766566. We acknowledge support from the Ministry of Education of the Czech Republic Grants LM2015087 and LNSM-LNSpin, and the Grant Agency of the Czech Republic Grant 14-37427G. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the programme 'Projects of Large Research, Development, and Innovations Infrastructures' (CESNET LM2015042) is greatly appreciated. Y.M. acknowledges funding from the German Research Foundation (Deutsche Forschungsgemeinschaft, Grant MO 1731/5-1). B.Y. acknowledges the support of the Ruth and Herman Albert Scholars Program for New Scientists at Weizmann Institute of Science, Israel. A.H.M. was supported by SHINES, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award SC0012670, Army Research Office (ARO) under Contract No. W911NF-15-1-0561:P00001, and by Welch Foundation Grant TBF1473.