Stable, large-volume, non-equilibrium plasmas, called Atmospheric-Pressure Glow Discharges (APGD), are emerging as an important new class of glow discharges with several potential applications in materials processing. These discharges operate in a previously inaccessible regime of the plasma parameter space, where properties resemble low-pressure glow plasmas but at significantly higher (atmospheric) pressures. Recently, several investigators have reported the generation of large-volume APGD and uses of APGD in the processing of materials. However, there exists no clear explanation of the structure of these discharges and the reasons for their stability. This paper reports detailed one-dimensional model-based investigation of a capacitively coupled APGD. The paper will discuss the structure of these highly collisional, non-equilibrium plasmas and the chemical nature of these discharges. Model predictions of the stability boundaries of the discharge will be reported. Results show that for certain operating conditions and working gas compositions, stable operating regimes between breakdown and arcing are obtained. Model predictions for discharge V-I characteristics and the stability boundaries are compared to experimental results reported in the literature. @FootnoteText@ This work is supported by a NSF-CAREER Award.