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I shall begin by reviewing what is known about the automorphisms of various complexesÌýnaturally associated to mapping class groups of surfaces, $Aut(F)$ and $Out(F)$, drawing analogies with theÌýclassical case of SL(n,Z). There are various rigidityÌýresults concerning the automorphism groupsÌýof these complexes, which can be viewed as analogues and extensions of the fundamental theorem ofÌýprojective geometry. In some cases these lead to algebraic rigidity results describing the abstractÌýcommensurators of the groups and various of their natural subgroups (eg the Torelli groups). The talk willÌýbe directed towards outliningÌýa proof of the following theorems: for $n\ge 3$ the action of $Aut(F_n)$ onÌýthe complex $FF_n$ of free factors gives anÌýisomorphism $Aut(F_n)\to Aut(FF_n)$; and the abstract commensuratorÌýof $Aut(F_n)$ is $Aut(F_n)$. The first result is joint work withÌýMladen Bestvina and the second is with Ric Wade.