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Cheese is classified by its ripening process (how it is aged) or lack thereof. Within these broader categories, cheese can be classified by its origins, its age (and, consequently, its degree of softness or hardness), its milk(s), and whether or not the milk has been pasteurized.
Different cheese making traditions will have slightly different classification systems due to the use of different methods. Here is an attempt at a fairly general system of classification based on ripening process:
What is how is cheese classified is