  We derive the complete set of evolutions of chirality-odd twist-3
fragmentation functions at one-loop level. There are totally nine real twist-3
fragmentation functions, among which seven are independent. The
renormalization-scale dependence of the nine functions has an important
implication for studies of single transverse-spin asymmetries. We find that the
evolutions of the three complex fragmentation functions defined by
quark-gluon-quark operator are mixed with themselves. There is no mixing with
the fragmentation functions defined only with bilinear quark field operators.
In the large-$N_c$ limit the evolutions of the three complex fragmentation
functions are simplified and reduced to six homogenous equations.
