This thesis aims to study some algebraic aspects of the deformation quantization problem. On one hand, we consider formality for free algebras and the Lie algebra so(3), and on the other hand we study deformation quantization for Hom-algebraic structures. Following Kontsevich's formality result in 1997 for symmetric algebras, we study in the first part free algebras, which are a particular case of envelopping algebras, and show that there is no formality, except for the trivial cases. We also show that there is no formality for the Lie algebra so(3). The tools used are homological ones. We compute the cohomology of these algebras and proceed to the construction of the L-infinity-quasi-isomorphism between the differential graded Lie algebra of the Hochschild cochains endowed with the Gerstenhaber bracket and the cohomology algebra endowed with the Schouten bracket. In the second part of this work, we use a principle of deformation by twist for Hom-algebraic structures, to construct new structures of the same type, or to deform a classical structure in the corresponding Hom-structure using an algebra morphism. In particular, we apply this method to Poisson structures and Moyal-Weyl star-products. Besides, we establish a correspondance between enveloping algebras of Hom-Lie algebras endowed with a Hom-coPoisson structure and Hom-Lie bialgebras.
