Here is a long article by Hazewinkel and a discussion on the nLab.

The functor of taking big Witt vectors is right adjoint to the forgetful functor from lambda-rings to commutative rings. Lambda rings appear in topological K-theory and representation theory, because vector bundles (and representations of groups) have exterior powers (these are the lambda operations).

W(Z) is apparently universal in several settings, such as symmetric function theory. Borger has a proposal for the category of schemes over the field with one element involving W(Z)-schemes.

Here is a long article by Hazewinkel and a discussion on the nLab.

The functor of taking big Witt vectors is right adjoint to the forgetful functor from lambda-rings to commutative rings. Lambda rings appear in topological K-theory and representation theory, because vector bundles (and representations of groups) have exterior powers (these are the lambda operations).

W(Z) is apparently universal in several settings, such as symmetric function theory (where I believe it is K₀(finite sets)). Borger has a proposal for the category of schemes over the field with one element involving W(Z)-schemes.