In this manuscript, we will discuss the notion of curved momentum space,
as it arises in the discussion of noncommutative or doubly special relativity theories.
We will illustrate it with two simple examples, the Casimir effect in anti-Snyder space
and the introduction of fermions in doubly special relativity.
We will point out the existence of intriguing results,
which suggest nontrivial connections with spectral geometry and Hopf algebras.