Abstract: The origin of deformation quantization goes back to as far as 1969 in its purely algebraic form. When applied this construction to the algebra $C^{\infty}(M)$ of smooth complex valued functions on a manifold $M$ , if exists, one obtains a quantization,making the space $C^{\infty}(M)$ noncommutative. Roughly speaking, the construction proceeds as follows: using the algebra $C^{\infty}(M)$ of complex valued smooth functions on $M$, one defines a new product $\star$ depending on some formal quantization parameter $\hbar$.This new product is viewed as formal power series in $\hbar$,thus defining a new algebra $C^{\infty}(M)[[\hbar ]]$ over the ring $\mathbb{C}[[\hbar]]$. An example of such a product called Weyl-Moyal product on $\mathbb{R}^{N}$ arises naturally from its Poisson structure. Under any new multiplication, $\frac{f\star g -g\star f}{\hbar}\vert_{\hbar\longrightarrow 0} = \{f,g\}$. In fact, M. Kontsevich proved that if $M$ has a Poisson bracket, then $M$ admits a nontrivial deformation quantization.I will sketch the proof of Kontsevich in the simplest case $M = \mathbb{R}^{n}$. As much as time is allotted, I will give as many applications of Kontsevich celebrated result as possible.