Abstract: We identify the $2$-dimensional torus $\mathbb{T}^2$ with $(\mathbb{R}/\mathbb{Z})^2$, and we study the dynamics of the continuous linear and quadratic polynomial differential systems on the torus $(\mathbb{R}/\mathbb{Z})^2$. The linear systems depend on two parameters, while the quadratic ones depend on five parameters. In particular, we characterize all the local phase portraits of their equilibrium points, we study their limit cycles,... Our final objective is to obtain the global phase portraits of these differential systems. This is a joint work with Ali Bakhshalizadeh.