Numerical study of the natural-convection flow and heat transfer in a dome-shaped, heat-generating, porous enclosure is considered. The general conic equation for the top dome is used to consider various conical top sections such as circular, elliptical, parabolic, and hyperbolic. The individual effect of fluid Rayleigh, Darcy, and heat-generating parameters on flow patterns and heat transfer rates are analyzed and presented. The predicted results show that the heat-generating parameter has the most significant contribution toward the growth of bicellular core flow. Moreover, there is significant change in temperature distribution in comparison to rectangular enclosures, due to the existence of the domed-shape top adiabatic cover. The results also show that, regardless of Darcy and Rayleigh values, a flat adiabatic top cover tends to yield the highest value of Nusselt number, followed by circular, elliptical, parabolic, and hyperbolic top covers, respectively.