In the twentieth century we use two words for that which in the Middle Ages were called universals. Instead of saying that every common name in the vocabulary of English or other European language signifies a universal, we say that it signifies a class or kind. This naming holds not only for nouns but also for adjectives and the participles of verbs.
The obvious question is, how do such universal -- classes or kinds -- exist? Individuals, particular things, or events we perceive, imagine, or remember exist in the physical world. They are signified by all the proper noun we use or when, in the place of proper noun such as "George Washington," we substitute a definite description, such as "the first President of the United States."
The answer to the question posed is, first, negative. Classes or kinds do not have real existence, the kind of existence that you and I have. They do not have mental existence -- the existence in our minds of cognitive mental content (which in modern times we misuse the word "ideas" to designate).
Positively, then, we must say that they have objective or intentional existence. (See Being.) They exist as objects of the mind, not as items in the mind. They exist for any two human beings who have before their minds an object that they can discuss.
In modern times, in the writing of Bishop George Berkeley and David Hume, the objective or intentional existence of kinds or classes is denied; and also denied is the mental existence of what Berkeley and Hume called "abstract ideas." For them, only words have general significance, applicable to two or more individuals that have some characteristic or trait in common. This error on their part is called "nominalism." (See Nominalism.)