  We describe the theoretical advances that influenced the experimental
creation of vibrationally and translationally cold polar $^{40}$K$^{87}$Rb
molecules \cite{nphys08,science08}. Cold molecules were created from
very-weakly bound molecules formed by magnetic field sweeps near a Feshbach
resonance in collisions of ultra-cold $^{40}$K and $^{87}$Rb atoms. Our
analysis include the multi-channel bound-state calculations of the hyperfine
and Zeeman mixed X$^1\Sigma^+$ and a$^3\Sigma^+$ vibrational levels. We find
excellent agreement with the hyperfine structure observed in experimental data.
In addition, we studied the spin-orbit mixing in the intermediate state of the
Raman transition. This allowed us to investigate its effect on the
vibrationally-averaged transition dipole moment to the lowest ro-vibrational
level of the X$^1\Sigma^+$ state. Finally, we obtained an estimate of the
polarizability of the initial and final ro-vibrational states of the Raman
transition near frequencies relevant for optical trapping of the molecules.
