Let G be a complex simple direct limit group, specifically SL(∞; C) , SO(∞; C) or Sp(∞; C). Let F be a (generalized) flag in C∞. If G is SO(∞; C) or Sp(∞; C) we suppose further that F is isotropic. Let Z denote the corresponding flag manifold; thus Z= G/ Q where Q is a parabolic subgroup of G. In a recent paper (Penkov and Wolf in Real group orbits on flag ind-varieties of SL∞(C) , to appear in Proceedings in Mathematics and Statistics) we studied real forms G0 of G and properties of their orbits on Z. Here we concentrate on open G0-orbits D⊂ Z. When G0 is of hermitian type we work out the complete G0-orbit structure of flag manifolds dual to the bounded symmetric domain for G0. Then we develop the structure of the corresponding cycle spaces MD. Finally we study the real and quaternionic analogs of these theories. All this extends results from the finite-dimensional cases on the structure of hermitian symmetric spaces and cycle spaces (in chronological order: Wolf in Bull Am Math Soc 75:1121–1237, 1969; Wolf et al. in Ann Math 105:397–448, 1977; Wolf in Ann Math 136:541–555, 1992; Wolf in Compact subvarieties in flag domains, 1994; Wolf and Zierau in Math Ann 316:529–545, 2000; Huckleberry et al. in Journal für die reine und angewandte Mathematik 2001:171–208, 2001; Huckleberry and Wolf in Cycle spaces of real forms of SLn(C) , Springer, New York, pp 111–133, 2002; Wolf and Zierau in J Lie Theory 13:189–191, 2003; Huckleberry and Wolf in Ann Scuola Norm Sup Pisa Cl Sci (5) 9:573-580, 2010).