According to Wikipedia,
The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.
I am aware of Russell's Paradox, which explains why not everything is a set, but how can we show the collection of all groups is a proper class?