We have studied interspecies scattering in an ultracold mixture of Rb-87 and Cs-133 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 cataloged. Magnetic field modulation spectroscopy was used to observe molecular states bound by up to 2.5 MHz x h. We have created RbCs Feshbach molecules using two of the resonances. Magnetic moment spectroscopy along the magnetoassociation 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 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 wave functions 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.