I am interested in the relation between Bousfield localizations of model categories and localizations of $(\infty,1)$-categories.
According to Hirschhorn's book we can form the left Bousfield localization of a left proper cellular model category along any set of maps. According to Lurie's book we can form the (left) localization of a presentable $\infty$-category along a small collection of maps. How are those two results related? Is the simplicial nerve of a left proper, cellular model category a presentable quasi-category?
Furthermore we also know that we can form the right Bousfield localization of a right proper cellular model category along any set of objects. Is there an analogous theory of right localizations on $\infty$-categories?
EDIT: It has been pointed out that combinatorial model categories correspond exactly to presentable $\infty$-categories and left Bousfield localizations exist for every left proper, simplicial, combinatorial model category and correspond exactly to the localizations of the corresponding presentable $\infty$-category.
This still leaves the question if there is a similar theory and an existence theorem for right localizations of $\infty$-categories corresponding to the theory of right Bousfield localizations of something like right proper, simplicial, cellular model categories.