The future use of the CERN SPS accelerator as injector for the Large Hadron Collider, LHC, and the possible use of the SPS as a neutrino source for the Gran Sasso experiment are pushing the maximum intensity requirements of the accelerator much higher than achieved up to now. At the same time the requirements on beam quality are becoming far more stringent. The SPS machine, built in the 70’s, is not a “smooth” machine. It contains many discontinuities in vacuum chamber cross-section and many cavity-like objects, as well as the 5 separate RF systems at present installed. All these lead to a high impedance, seen by the beam, spread over a wide frequency range. As a result there is a constant fight against instabilities, both single and multi bunch, as the intensity increases. A program of studies is under way in the SPS to identify, reduce, and remove where possible the sources of these impedances. ©1999 American Institute of Physics.