  Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $G_K =
\mathrm{Gal}(\bar{\mathbf{Q}}_p/K)$. There is a very useful classification of
$p$-adic representations of $G_K$ in terms of cyclotomic
$(\varphi,\Gamma)$-modules (cyclotomic means that $\Gamma={\rm
Gal}(K_\infty/K)$ where $K_\infty$ is the cyclotomic extension of $K$). One
particularly convenient feature of the cyclotomic theory is the fact that any
$(\varphi,\Gamma)$-module is overconvergent.
  Questions pertaining to the $p$-adic local Langlands correspondence lead us
to ask for a generalization of the theory of $(\varphi,\Gamma)$-modules, with
the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie
extension $K_\infty / K$. It is not clear what shape such a generalization
should have in general. Even in the case where we have such a generalization,
namely the case of a Lubin-Tate extension, most $(\varphi,\Gamma)$-modules fail
to be overconvergent.
  In this article, we develop an approach that gives a solution to both
problems at the same time, by considering the locally analytic vectors for the
action of $\Gamma$ inside some big modules defined using Fontaine's rings of
periods. We show that, in the cyclotomic case, we recover the ususal
overconvergent $(\varphi,\Gamma)$-modules. In the Lubin-Tate case, we can
prove, as an application of our theory, a folklore conjecture in the field
stating that $(\varphi,\Gamma)$-modules attached to $F$-analytic
representations are overconvergent.
