Detachment folds seem conceptually the simplest of all fault-related folds, yet they continue to offer substantial and rich challenges that are symptomatic of fault-related folding in general. The standard picture of detachment folds goes back at least to August Buxtorf's widely reproduced 1916 regional cross sections of the Swiss Jura in which he interpreted the Jura box folds to be regionally detached from the basement along a weak layer of Triassic evaporites. Most theoretical and conceptual models of detachment folding have followed Buxtorf's model with a twofold mechanical stratigraphy composed of a competent flexural lid conserving bed length overlying a weak basal detachment layer that conserves only volume. However, this standard model of detachment folding has a fundamental and classic problem from a balancing perspective, which is outlined in this chapter. Furthermore, the classic Jura box folds drawn by Buxtorf were in fact not at all constrained by subsurface data, except along the Grenchenberg railroad tunnel where a surprising and much more complex structure was encountered, involving a strongly folded thrust fault that was intersected three times in the 8.5-km (5.3-mi)-long tunnel. Hans Laubscher argued that such complexly folded thrust faults are in fact typical of many Jura box folds. Analogous structures have been observed in Canada, Mexico, and Spain. I argue here on the basis of mass-balance considerations that detachment folds with folded thrusts are one of several theoretically expected modes of detachment folding. More broadly, the classical models of detachment folding assume a closed-system behavior in which all shortening is locally consumed in the fold. These classical models include (1) local flow of a weak basal layer, for example, salt (Wiltschko and Chapple mechanism) and (2) pure-shear detachment folding (Groshong and Epard mechanism). Here we discuss several additional models of detachment folding, including open-system behavior with (3) far-field flow of the weak basal layer, (4) roof detachments above the basal layer, and (5) roof ramps above the basal layer such as the folded thrust fault of the Grenchenberg anticline. In this final mechanism, the flexural lid or other more competent stratigraphic intervals shorten by a combination of fault slip and flexure, whereas the less competent intervals, including the basal layer, shorten by flow. All of these mechanisms are end members capable of existing in combination within actual folds; several examples combine two or three of these mechanisms.