This thesis describes an experiment to produce the first laser cooled mixtures of atoms
and polar molecules in magneto-optical and magnetic quadrupole traps. Inelastic collision
processes between CaF molecules and Rb atoms are studied in both trapping
arrangements. When molecules occupy the first rotationally excited state it is found that
loss from either trap occurs at near the universal rate due to collisions with co-trapped
atoms. When the molecular sample is transferred to the ground rotational state the loss rate is suppressed by at least an order of magnitude. The experimental results are compared to a quantum scattering model based on quantum defect theory and the results seem
to be encouraging for observing a high ratio of elastic to inelastic collision rates in this
system. In addition, improvements to the experimental apparatus are presented which
led to the first measurements of the elastic cross-section for collisions between CaF and
Rb. Although at the currently attainable number densities no thermalization is observed,
an upper bound on the value of the cross section is placed.