  This thesis is devoted to the study of geometric properties of affine
algebraic varieties endowed with an action of an algebraic torus. It comes from
three preprints which correspond to the indicated points (1), (2), (3). Let $X$
be an affine variety equipped with an action of the algebraic torus
$\mathbb{T}$. The complexity of the $\mathbb{T}$-action on $X$ is the
codimension of the general $\mathbb{T}$-orbits. Under the assumption of
normality and when the ground field is algebraically closed of characteristic
0, the variety $X$ admits a combinatorial description in terms of convex
geometry. This description obtained by Altmann and Hausen in the year 2006
generalizes the classical one for toric varieties. Our purpose is to
investigate new problems on the algebraic and geometric properties of the
variety $X$ when the $\mathbb{T}$-action on $X$ is of complexity 1. (1) In the
first part, a result gives an effective method to determine the integral
closure of any affine variety defined over an algebraically field of
characteristic 0 with a $\mathbb{T}$-action of complexity 1 in terms of the
combinatorial description of Altmann-Hausen. (2) The calculations of the first
part suggest a proof of the validity of the presentation of Altmann-Hausen in
the case of complexity 1 over an arbitrary ground field. This is done in the
second part. (3) In the third part, when the base field is perfect, we classify
all the actions of the additive group on $X$ that are normalized by the
$\mathbb{T}$-action of complexity 1. This classification generalizes classical
works of Flenner and Zaidenberg in the surface case and of Liendo when the base
field is algebraically closed of characteristic 0.
