Abstract: for problems involving the interaction of large-scale geophysical fluid dynamics and smaller-scale deep convection, I will describe a set of methods in which the convection in a small horizontal domain is modeled explicitly while larger scales are parameterized, i.e., represented in a highly simplified way without explicitly representing the flow outside the small domain. The oldest and most-used method of this type is the weak temperature gradient (WTG) approximation, which is appropriate for a set of problems in tropical weather and climate dynamics. I will review and summarize our work on WTG, and then proceed to describe a new approach of this type, appropriate in the extratropics, which we call "column quasi-geostrophy" (CQG), as it uses the quasi-geostrophic omega equation to model the large-scale vertical velocity. As in WTG, the large-scale vertical velocity is influenced by diabatic heating, but it also contains dry adiabatic terms, related to potential vorticity dynamics, which are absent in WTG. These dry dynamic terms are specified in CQG, while the convection is explicitly modeled and its diabatic heating allowed to influence large-scale vertical motion. We apply CQG to study the dynamics of extreme precipitation events, including those leading to floods in 2010 in Pakistan and 2015 in Texas. We use the latter case as a vehicle to diagnose the processes leading to changes in precipitation extremes in a warming climate, especially feedbacks between moisture, latent heating, and large-scale ascent.