  The subject of this Master 2 thesis is the study of the
Andr\'e-Chudnovsky-Katz theorem on the structure of the solution of the nonzero
differential equation of minimal order with coefficients in
$\overline{\mathbb{Q}}(z)$ satisfied by a $G$-function. We begin by presenting
the theory of globally nilpotent differential operators, of which the main
result is the Katz theorem, which states that they are fuchsian with rational
exponents. We then give a full proof of the Chudnovsky theorem implying that
the minimal nonzero differential operator with coefficients in
$\overline{\mathbb{Q}}(z)$ of a $G$-function satisfies a moderate growth
condition on some denominators called the Galochkin condition. We finally
outline the proof of the Andr\'e-Bombieri theorem establishing the equivalence
between the Galochkin condition and the Bombieri condition, which implies the
global nilpotence. This allows us to prove the main point of the
Andr\'e-Chudnovsky-Katz theorem.
