This thesis consists of three articles relatedto Sen theory and its applications. In the first article, we let $K$ be a finite extension of $\Q_{p}$and let $\Gamma$ be the Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a classification of $p$-adic representations of $\Gal(\overline{K}/K)$ in terms of $(\varphi,\Gamma)$-modules. A useful aspect of this classification is Berger's dictionary which expresses invariants coming from $p$-adic Hodge theory in terms of these $(\varphi,\Gamma)$-modules. We use the theory of locally analytic vectors to generalize this dictionary to the setting where $\Gamma$ is the Galois group of a Lubin-Tate extension of $K$. As an application, we show that if $F$ is a totally real number field and $v$ is a place of $F$ lying above $p$, then the $p$-adic representation of $\Gal(\overline{F}_{v}/F_{v})$ associated to a finite slope overconvergent Hilbert eigenform which is $F_{v}$-analytic up to a twist is Lubin-Tate trianguline. Furthermore, we determine a triangulation in terms of a Hecke eigenvalue at $v$. This generalizes results in the case $F=\Q$ obtained previously by Chenevier, Colmez and Kisin. In the second article, we develop a version of Sen theory for equivariantvector bundles on the Fargues-Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of $(\varphi,\Gamma)$-modules in the cyclotomic case then recovers the Cherbonnier-Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles. Using the $p$-adic monodromy theorem, we show that each locally analytic vector bundle $\mathcal{E}$ has a canonical differential equation for which the space of solutions has full rank. As a consequence, $\mathcal{E}$ and its sheaf of solutions $\Sol(\mathcal{E})$ are in a natural correspondence, which gives a geometric interpretation of a result of Berger on $(\varphi,\Gamma)$-modules. In particular, if $V$ is a de Rham Galois representation, its associated filtered $(\varphi,N,G_{K})$-module is realized as the space of global solutions to the differential equation. A key to our approach is a vanishing result for the higher locally analytic vectors of representations satisfying the Tate-Sen formalism, which is also of independent interest. Finally, in the third article, we prove a conjecture of Emerton, Geeand Hellmann concerning the overconvergence of étale $(\varphi,\Gamma)$-modules in families parametrized by topologically finite type $\Z_{p}$-algebras. As a consequence, we deduce the existence of a natural map from the rigid fiber of the Emerton-Gee stack to the rigid analytic stack of $(\varphi,\Gamma)$-modules.