  This dissertation is an exposition of Kontsevich's proof of the formality
theorem and the classification of deformation quantisation on a Poisson
manifold. We begin with an account of the physical background and introduce the
Weyl-Moyal product as the first example. Then we develop the deformation theory
via differential graded Lie algebras and $\mathrm{L}_\infty$-algebras, which
allows us to reformulate the classification of deformation quantisation as the
existence of a $\mathrm{L}_\infty$-quasi-isomorphism between two differential
graded Lie algebras, known as the formality theorem. Next we present
Kontsevich's proof of the formality theorem in $\mathbb{R}^d$ and his
construction of the star product. We conclude with a brief discussion of the
globalisation of Kontsevich star product on Poisson manifolds.
