A recent result of Martin Bridson and Karen Vogtmann for automorphism groups of free groups shows that there is no nontrivial morphism from the group Aut~$F_n$ into Aut~$F_m$ when $n>m>1$. In this talk, we exploit some well-known facts about Dehn twists, together with a specific twist description of a torsion element of maximal order in the mapping class group ${\rm Mod}_g$, to show that this same result holds true for ${\rm Mod}_g$; any homomorphism $\varphi:{\rm Mod}_g\to{\rm Mod}_h$ between mapping class groups of closed orientable surfaces with distinct genera $g>h$ is trivial if $g\geq 3$ and has finite image for all $g\geq 1$. Some implications are drawn for more general homomorphs of mapping class groups.