We present first-principles results on the structural, electronic, and magnetic properties of a new family of twodimensional antiferromagnetic (AFM) manganese chalcogenides, namely, monolayer MnX and Janus XMnY (X, Y = S, Se, Te), among which monolayer MnSe was recently synthesized in experiments [Aapro et al., ACS Nano 15, 13794 (2021)]. By carrying out calculations of the phonon dispersion and ab initio molecular dynamics simulations, we first confirmed that these systems, characterized by an unconventional strongly-coupled-bilayer atomic structure [consisting of Mn atoms buckled to chalcogens forming top and bottom ferromagnetic (FM) planes with antiparallel spin orientation], are dynamically and thermally stable. The analysis of the magnetic properties shows that these materials have robust AFM order, retaining a much lower energy than the FM state even under strain. Our electronic structure calculations reveal that pristine MnX and their Janus counterparts are indirect-gap semiconductors, covering a wide energy range and displaying tunable band gaps by the application of biaxial tensile and compressive strain. Interestingly, owing to the absence of inversion and time-reversal symmetry, and the presence of an asymmetrical potential in the out-of-plane direction, Janus XMnY become spin-split gapped systems, presenting a rich physics yet to be explored. Our findings provide insights into this physics and highlight the potential for these two-dimensional manganese chalcogenides in AFM spintronics.