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We prove that the mapping class group $\Gamma_{g,n}$ for surfaces of negativeEuler characteristic has a cofinite universal space $\E$ for proper actions(the resulting quotient is a finite $CW$-complex). The approach is to constructa truncated Teichmueller space $\T_{g,n}(\epsilon)$ by introducing a lowerbound for the length of shortest closed geodesics and showing that$\T_{g,n}(\epsilon)$ is a $\Gamma_{g,n}$ equivariant deformation retract of theTeichmueller space $\T_{g, n}$. The existence of such a cofinite universalspace is important in the study of the cohomology of the group $\gag$. As anapplication, we note that there are only finitely many conjugacy classes offinite subgroups of $\Gamma_{g,n}$. Another application is that the rationalNovikov conjecture in K-theory holds for $\Gamma_{g,n}$.

A cofinite universal space for proper actions for mapping class groups