Two methods for assessing the hydraulic criticality of an observed or modeled overflow are discussed. The methods are valid for single-layer deep flows with arbitrary potential vorticity and cross section. The first method is based on a purely steady view in which the flow at a given section is divided up into a group of “streamtubes.” A hydraulic analysis requires an extension of Gill’s functional formulation to systems with many degrees of freedom. The general form of the critical condition and associated compatibility condition for such a system are derived and applied to the streamtube model. As an aside, it is shown by example that Gill’s original critical condition can fail to capture all possible critical states, but that this problem is fixed when the multivariable approach is used. It is also shown how Gill’s method can be applied to certain dispersive or dissipative systems. The second method of assessing criticality involves direct calculation of linear, long-wave speeds using a time-dependent version of the streamtube model. This approach turns out to be better suited to the analysis of geophysical datasets. The significance of the local Froude number F is discussed. It is argued that F must take on the value unity at some point across a critical section.