  We have studied interspecies scattering in an ultracold mixture of $^{87}$Rb
and $^{133}$Cs atoms, both in their lowest-energy spin states. The three-body
loss signatures of 30 incoming s- and p-wave magnetic Feshbach resonances over
the range 0 to 667 G have been catalogued. Magnetic field modulation
spectroscopy was used to observe molecular states bound by up to 2.5 MHz$\times
h$. Magnetic moment spectroscopy along the magneto-association pathway from 197
to 182 G gives results consistent with the observed and calculated dependence
of the binding energy on magnetic field strength. We have created RbCs Feshbach
molecules using two of the resonances. We have set up a coupled-channel model
of the interaction and have used direct least-squares fitting to refine its
parameters to fit the experimental results from the Feshbach molecules, in
addition to the Feshbach resonance positions and the spectroscopic results for
deeply bound levels. The final model gives a good description of all the
experimental results and predicts a large resonance near 790 G, which may be
useful for tuning the interspecies scattering properties. Quantum numbers and
vibrational wavefunctions from the model can also be used to choose optimal
initial states of Feshbach molecules for their transfer to the rovibronic
ground state using stimulated Raman adiabatic passage (STIRAP).
