From a mechanical point of view, human skin appears as a layered composite containing the stiff thin cover layer presented by the stratum corneum, below which are the more compliant layers of viable epidermis and dermis and further below the much more compliant adjacent layer of subcutaneous white adipose tissue (sWAT). Upon exposure to a strain, such a multi-layer system demonstrates structural instabilities in its stiffer layers, which in its simplest form is the wrinkling. These instabilities appear hierarchically when the mechanical strain in the skin exceeds some critical values. Their appearance is mainly dependent on the mismatch in mechanical properties between adjacent skin layers or between the skin and sWAT, on the adhesive strength and thickness ratios between the layers, on their bending and tensile stiffness as well as on the value of the stress existing in single layers. Gradual reduction of elastic fibers in aging significantly reduces the skin's ability to bend, prompting an up to 4-fold reduction of its stability against wrinkling, thereby explaining the role of these fibers in skin aging. While chronological and extrinsic aging differently modify these parameters, they lead to the same end result, reducing the critical strain required for the onset of instabilities. Comparing of mechanical properties of the skin presented as a bi-, tri- or tetra-layer structure demonstrates the particular importance of the papillary dermis in skin aging and provides the arguments to consider the undulations on the dermal-epidermal and dermal-sWAT interfaces as the result of mechanical bifurcation, leading to structural instabilities inside of the skin. According to this model, anti-aging strategies should focus not as much on the reinforcement of the dermis, but rather aim to treat the elastic mismatch between different adjacent layers in the skin and sWAT as well as the adhesion between these layers.