On the relation between $\infty$-categories and model categories

We will report on joint work with Steffen Sagave. First we review the notions of model categories and $\infty$-categories which are both frameworks to deal with homotopy coherence and higher structures. The relation between the two is well understood by now thanks to work of Dugger, Lurie and Joyal. We will explain the relation and if time permits some applications. Our main contribution is to give a symmetric monoidal variant of this story: we prove that symmetric monoidal $\infty$-categories and some sort of symmetric monoidal $\infty$-catgories are essentially the same. This resolves an open question (asked e.g. by Lurie)  and we will explain how to use such a result.