It is a theorem of Gopal Prasad (which I hope I am not misquoting...) that lattices in higher rank linear semi-simple Lie groups have finite outer automorphism groups. Is there some other reasonable class of groups with this finiteness property?
Here is a result of Fr\'ed\'eric Paulin: a hyperbolic group with Kazhdan's property (T), has a finite outer automorphism group. See Outer automorphisms of hyperbolic groups and small actions on R-trees. Arboreal group theory (Berkeley, CA, 1988), 331–343, Math. Sci. Res. Inst. Publ., 19, Springer, New York, 1991.
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The theorem Igor Belegradek mentions is a more general form of the theorem (due collectively to Bestvina, Feighn, Paulin, and Rips) that a one-ended hyperbolic group has infinite outer automorphism group only if it splits over a two-ended subgroup. See
M. Bestvina, M. Feighn, Stable actions of groups on real trees. Invent. Math. 121 (1995), no. 2, 287-321.