  The dissertation deals with noncommutative field theories, namely field
theories compatible with the existence of a minimal (quantum gravity) length
scale. Two families of quantum spacetime are considered. One is characterized
by semisimple Lie algebras of coordinates. The other is characterized by
solvable Lie algebras. The explicit construction of corresponding Weyl-like
star products is given. We then focus on two specific examples of quantum space
and study the quantum properties of various models of scalar field theory with
quartic interactions built on them. The corresponding 2-point and 4-point
functions are computed at one-loop and the UV/IR mixing is discussed. The first
quantum spacetime considered is known as $\mathbb{R}^3_\theta$ which is a
deformation of $\mathbb{R}^3$ with $\mathfrak{su}(2)$ noncommutativity. In this
case, the one-loop 2-point function is found to be finite with the deformation
parameter playing the role of a cutoff. The second quantum space is known as
$\kappa$-Minkowski. In this case, the action functional is required
$\kappa$-Poincar\'e invariant and various kinetic operators are considered. In
one case, we find that the one-loop 2-point function has milder UV divergence
than in the commutative case and that the one-loop 4-point function is finite.
The renormalization properties are discussed. Besides, the loss of cyclicity of
the Lebesgue integral (which results from the $\kappa$-Poincar\'e invariance of
the action functional) is interpreted as reflecting the occurrence of KMS
condition at the level of the algebra of fields modelling $\kappa$-Minkowski.
This interpretation sheds new light on $\kappa$-deformation-based quantum
gravity models, and solved a 25 years old problem in the study of
$\kappa$-Poincar\'e invariant quantum field theories. Possible extensions of
this work are finally discussed.
