I have discovered cosmetic cold-wax hair removal (depilatory) compositions suitable for site-specific face or body applications with the following attributes: (1) A pre-heating or microwave step is not necessary, (2) Complete hair removal is achieved in a single application on a specific site or area, (3), No (or minimal) skin irritation is experienced, (4) Bond to hair preferentially, and not to skin, (5) Provide lubricity and emolliency to skin, (6) Can be applied either directly on the skin or, preferably, on a piece of precut fabric, plastic or paper, which is then applied to specific site where hair removal is desired. Alternatively, dispensing applicators can also be used that can be filled for delivery at the specific site where depilation is desired. (7) Require minimal clean-up with water or common make-up removal compositions, and (9) Additional skin and hair beneficial ingredients, such as hair growth retardants, anti-irritants, topical pain relief agents, antioxidants, skin soothing agents, skin cooling agents, emollients, moisturizers, topical anesthetics, colorants, botanical extracts, fragrances, and such can be included.