A celebrated theorem of Tits asserts that every finitely generated linear group is either virtually solvable or contains a non-abelian free subgroup. The same dichotomy was later proved to hold for subgroups of surface mapping class groups (due to McCarthy and Ivanov) and Out(Fn), the outer automorphism of a rank n free group (due to Bestvina-Feighn-Handel); in these cases solvable subgroups are in fact known to be virtually abelian. Several strengthenings of this alternative have then been proved. The goal of the mini-course will be to present the following strengthening of the Bestvina-Feighn-Handel theorem: every non virtually abelian subgroup H of Out(Fn) is SQ-universal, that is, every countable group embeds in a quotient of H. This is proved by showing that H virtually surjects onto an acylindrically hyperbolic group. This is a joint work with Vincent Guirardel.