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We demonstrate, to our knowledge, the first bright circularly polarized high-harmonic beams in the soft X-ray region of the electromagnetic spectrum, and use them to implement X-ray magnetic circular dichroism measurements in a tabletop-scale setup. Using counterrotating circularly polarized laser fields at 1.3 and 0.79 µm, we generate circularly polarized harmonics with photon energies exceeding 160 eV. The harmonic spectra emerge as a sequence of closely spaced pairs of left and right circularly polarized peaks, with energies determined by conservation of energy and spin angular momentum. We explain the single-atom and macroscopic physics by identifying the dominant electron quantum trajectories and optimal phase-matching conditions. The first advanced phase-matched propagation simulations for circularly polarized harmonics reveal the influence of the finite phase-matching temporal window on the spectrum, as well as the unique polarization-shaped attosecond pulse train. Finally, we use, to our knowledge, the first tabletop X-ray magnetic circular dichroism measurements at the N4,5 absorption edges of Gd to validate the high degree of circularity, brightness, and stability of this light source. These results demonstrate the feasibility of manipulating the polarization, spectrum, and temporal shape of high harmonics in the soft X-ray region by manipulating the driving laser waveform. ; The authors thank Wilhelm Becker and Luis Plaja for useful discussions. Support for this work was provided by the Department of Energy (DOE) Office of Basic Energy Sciences X-Ray Scattering Program and the National Science Foundation (NSF) Physics Frontier Center Program Grant PHY-1125844 (to T.F., P.G., R.K., D.D.H., D.Z., C.G., F.J.D., C.A.M., C.W.H., J.L.E., K.M.D., C.C., T.P., A.B., H.C.K. and M.M.M.); NSF Graduate Research Fellowship DGE-1144083 (to J.L.E.); Marie Curie International Outgoing Fellowship within the European Union (EU) Seventh Framework Program for Research and Technological Development (2007–2013), under REA Grant 328334 (to C.H.-G.); Junta de Castilla y León Project SA116U13, UIC016 (to C.H.-G.); MINECO Grant FIS2013-44174-P (to C.H.-G.); US NSF Grants PHY- 1125844 and PHY-1068706 (to A.A.J.-B.); Deutsche Forschungsgemeinschaft Grant GR 4234/1-1 (to P.G.); Swedish Research Council (R.K. and P.M.O.); EU Seventh Framework Programme Grant 281043, FemtoSpin (to K.C. and P.M.O.); Czech Science Foundation Grant 15-08740Y (to K.C.); IT4Innovations Centre of Excellence Project CZ.1.05/1.1.00/02.0070 funded by the European Regional Development Fund and the national budget of the Czech Republic Project Large Research, Development and Innovations Infrastructures LM2011033 (to D.L.); US DOE, Office of Science, Office of Basic Energy Sciences Contract DE-SC0001805 (to O.G.S.); and US DOE Office of Basic Energy Sciences Award DE-SC0003678 (to E.E.F.). This work used the Janus supercomputer, which is supported by US NSF Award CNS-0821794 and the University of Colorado, Boulder.