  Systematic shell-model calculations are presented for odd-mass Cd isotopes
with $N=51-81$ utilizing a combination of a $G$-matrix interaction and a
semiempirical one. The excited energy spectra and electromagnetic transition
probabilities are compared with the recently available experimental data. We
have found that the observed quadrupole moments in the $11/2^-_1$ states that
linearly change with the neutron number are well accounted for by the dominance
of prolate shapes throughout the Cd isotope chain. We have also described the
properties of several isomeric states in neutron-rich $^{127-131}$In isotopes
that were recently observed in Jyv\"askyl\"a.
