The problem of the approximate parametrization of algebraic plane curves goes as follows: given a plane algebraic curve $\cal C$ (that is the perturbation of a rational plane curve) and a tolerance $\epsilon>0$, we want to find a new curve $\overline{\cal C}$, being rational, as well as a rational parametrization of it such that $\cal C$ is in the offset region of $\overline{\cal C}$, at certain small distance dependent on $\epsilon$, and vice-versa.

In this talk, we will give a brief overview of our results and on going on work on the problem.