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Mapping class groups of surfaces act properly on the corresponding Teichmuller spaces. This action is important for many applications. For example, the quotient gives the moduli spaces of Riemann surfaces, and the Teichmuller spaces provide explicit models for universal spaces for proper actions of the mapping class groups. One folklore problem concerns the existence of spines, i.e., equivariant deformation retracts of Teichmuller spaces, which are as small as possible. Mapping class groups share many properties with nonuniform arithmetic subgroups of semisimple Lie groups. The arithmetic groups act properly on corresponding symmetric spaces, and there is a corresponding problem on the existence of spines of optimal dimension. In this talk, I will describe several results on these problems about spines.