Closed reduced equations analogous to the quasi-geostrophic equations are derived in the extratropics for small Rossby numbers and vertical scales that are comparable to or much larger than horizontal scales. On these scales, significant vertical motions are permitted and found to couple to balanced geostrophic dynamics. In the equatorial regions, similar reduced equations are derived for meridional scales much larger than the vertical and zonal scales. These equations are derived by a systematic exploration of different aspect ratios, and Froude and buoyancy numbers, and offer advantages similar to the standard quasi-geostrophic equations for studies of smaller-scale processes and/or of the equatorial regions.