The authors study Sobolev classes of weakly differentiable mappings $f:{mathbb X}rightarrow {mathbb Y}$ between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in ${mathcal W}{1,n}({mathbb X}, ,, {mathbb Y}),$, $n=mbox{dim}, {mathbb X}$. The central themes being discussed are: