The Principle of Material Frame-Indifference in various areas of mechanics is critically reviewed from a basic theoretical standpoint. Modern continuum mechanics is considered along with statistical mechanics and turbulence in an effort to better understand this commonly used axiom. It is argued that Material Frame-Indifference is a restricted invariance that can be highly useful in the formulation of constitutive equations but must be applied with caution. Material Frame-Indifference applies, in a strong approximate sense, to most areas of continuum mechanics where there is a clear cut separation of scales so that the ratio of fluctuating to mean time scales is extremely small. While it breaks down for the three-dimensional case, it rigorously applies to Reynolds stress models in the limit of two-dimensional turbulence where an analogy is made between the Reynolds stress tensor and the non-Newtonian part of the stress tensor in the laminar flow of a non-Newtonian fluid. On the other hand, the general invariance group of constitutive equations that is universally valid is the extended Galilean group of transformations which includes arbitrary time-dependent translations of the spatial frame of reference; rotational frame-dependence then enters exclusively through the intrinsic spin tensor. In order to definitively address this issue it is necessary to establish what the invariance group is of solutions to the fluctuation dynamics from which constitutive equations are formally constructed. The implications of these results for future research in a variety of different fields in mechanics are thoroughly discussed. This article includes 52 references.