  Well-defined non-perturbative formulations of the physics of string theories,
sometimes with D-branes present, were identified over a decade ago, from a
careful study of double scaled matrix models. Following recent work which
recasts some of those old results in the context of type 0 string theory, a
study is made of a much larger family of models, which are proposed as type 0A
models of the entire superconformal minimal series coupled to gravity. This
gives many further examples of important physical phenomena, including
non-perturbative descriptions of transitions between D-branes and fluxes,
tachyon condensation, and holography. In particular, features of a large family
of non-perturbatively stable string equations are studied, and results are
extracted which pertain to type 0A string theory, with D-branes and fluxes, in
this large class of backgrounds. For the entire construction to work, large
parts of the spectrum of the supergravitationally dressed superconformal
minimal models and that of the gravitationally dressed bosonic conformal
minimal models must coincide, and it is shown how this happens. The example of
the super-dressed tricritical Ising model is studied in some detail.
