In English, possession is called out as a specific conjugation of a noun; "Fred's sandwich," "mens' clothing," "a butcher's knife," and pronouns have specific possessive forms, such as in "my cat." In many other languages, however, a possessive is often a specific form of a noun-formed adjective.
In Japanese, for example, you'd say, "Nikuya no naifu," "Fred-san no sandoitchi," "watashi no neko," and so on ("the knife of a butcher," "the sandwich of Fred," "the cat of me"). It's the same way that you'd form an adjective from a noun; "taiyō no iro" is the color of the sun. (In some cases the "no" can be left out, though; for example, one word for grey is "nezumi-iro" which literally means "mouse-colored.")
Spanish uses pretty much the same construct for possession by a noun; "chuchillo de carniceria" and "sándwich de Fred," for example. However, pronouns do have possessive forms; "mi gato," "sus zapatos," and so on. (One interesting twist with Spanish possessive pronouns is that they have two numbers associated with them — one for the subject, and one for the object. "Mis" is the first-person singular possession of a plural object, e.g. "mis zapatos" for "my shoes," whereas "nuestro" is the first-person plural possession of a singular object, e.g. "nuestro gato" for "our cat." The first-plural-person possessive pronouns also have to have gender agreement with the object; "nuestras camisas" for "our shirts.")
Of course, in English, it makes perfect sense to use these other languages' constructs, even if it usually sounds a bit more cumbersome. However, noun-as-adjective possession still makes sense sometimes; "the scent of a woman" can sound fairly romantic in the right context, whereas there aren't many times that "a woman's scent" wouldn't be just plain creepy.