  Let $K$ be a finite field extension of $Q_p$ and let $G_K$ be its absolute
Galois group. We construct the universal family of filtered $(\phi,N)$-modules,
or (more generally) the universal family of $(\phi,N)$-modules with a
Hodge-Pink lattice, and study its geometric properties. Building on this, we
construct the universal family of semi-stable $G_K$-representations in
$Q_p$-algebras. All these universal families are parametrized by moduli spaces
which are Artin stacks in schemes or in adic spaces locally of finite type over
$Q_p$ in the sense of Huber. This has conjectural applications to the $p$-adic
local Langlands program.
