Nonuniform liquid sheets are encountered in various industrial applications including radiation cooling in space (Chubb et al., 1994), and paper making (Soderberg and Alfredson, 1998). Many of the works cited below were motivated by applications in surface coating, fuel spray formation, nuclear safety, and other industrial processes. The spatial variation of sheet thicknesses in these applications is necessitated by the conservation of mass flow across the cross section perpendicular to the flow direction. For example, the thickness of a planar sheet of constant width must decrease in the flow direction due to the gravitational acceleration. Consequently the local Weber number, based on the local thickness and velocity, changes spatially. In particular We may be greater than one in part of the sheet and smaller than one in the rest. If one locally applies the concept of absolute and convective instability in a uniform sheet, then part of the sheet may experience convective instability, while the remaining part may experience absolute instability. Depending on the relative location of the regions of We > 1 and We < 1, one would expect different physical consequences to the entire flow. The objective of this chapter is to elucidate the effect of the spatial variation of We on the dynamics of sheet breakup by properly applying the concept developed in the previous chapter for a sheet of uniform thickness.