  For $1<p\leqslant \infty$, we study the complexity and the existence of
universal spaces for two classes of separable Banach spaces, denoted
$\textsf{A}_p$ and $\textsf{N}_p$, and related to asymptotic smoothness in
Banach spaces. We show that each of these classes is Borel in the class of
separable Banach spaces. Then we build small families of Banach spaces that are
both injectively and surjectively universal for these classes. Finally, we
prove the optimality of this universality result, by proving in particular that
none of these classes admits a universal space.
