  Neutrino Factories and Muon Colliders' cooling lattices require both high
gradient rf and strong focusing solenoids. Experiments have shown that there
may be serious problems operating rf in the required magnetic fields. The use
of high pressure gas to avoid these problems is discussed, including possible
loss problems from electron and ion production by the passage of an ionizing
beam. It is also noted that high pressure gas cannot be used in later stages of
cooling for a muon collider. Experimental observations using vacuum rf cavities
in magnetic fields are discussed, current published models of breakdown with
and without magnetic fields are summarized, and some of their predictions
compared with observations.
  A new theory of magnetic field dependent breakdown is presented. It is
proposed that electrons emitted by field emission on asperities on one side of
a cavity are focused by the magnetic field to the other side where they melt
the cavity surface in small spots. Metal is then electrostatically drawn from
the molten spots, becomes vaporized and ionized by field emission from the
remaining damage and cause breakdown. The theory is fitted to existing 805 MHz
data and predictions are made for performance at 201 MHz. The model predicts
breakdown gradients significantly below those specified for either the
International Scoping Study (ISS) Neutrino Factory or a Muon Collider.
  Possible solutions to these problems are discussed, including designs for
`magnetically insulated rf' in which the cavity walls are designed to be
parallel to a chosen magnetic field contour line and consequently damage from
field emission is suppressed. An experimental program to study these problems
and their possible solution is outlined.
