Fats are ubiquitous in biological membranes, foods, and many other commercial products. In these, they play essential roles in biological, nutritional, and physical functions. In this review, we focus on physical mechanical functions. The rheology of fats arises from the crystal network, which displays hierarchical structural levels from the molecular to the mesoscopic. Under linear deformations, the crystal network behaves as a viscoelastic solid with elasticity dictated by particle concentration and microstructural features as represented in fractal rheo-mechanical models. Under nonlinear deformations, the crystal network yields, showing a variety of nonlinear phenomena, i.e., softening, stiffening, thixotropy. These features largely contribute to functionality or performance as essentially all processing and end-uses of fatty materials involve large nonlinear deformations. Early work on rheology of fats gave hints of their nonlinear mechanical behavior, although in many cases the measured properties were empirical. In contrast, recent efforts from our group measured fundamental rheological functions using large amplitude oscillatory shear rheology. We demonstrate the ability of this technique to discern among the bulk functionality of bakery fats (all-purpose and lamination shortenings) based on well-defined rheological signatures that also relate to the fat structure. This technique has the potential to provide similar insights on other fatty systems and novel ideas for reformulation and design of alternative lipid-structuring materials.