  We study reaction-diffusion equations in one spatial dimension and with
general (space- or time-) inhomogeneous mixed bistable-ignition reactions. For
those satisfying a simple quantitative hypothesis, we prove existence and
uniqueness of transition fronts, as well as convergence of "typical" solutions
to the unique transition front (the existence part even extends to mixed
bistable-ignition-monostable reactions). These results also hold for all pure
ignition reactions without any other hypotheses, but not for all pure bistable
reactions. In fact, we find examples of either spatially or temporally periodic
pure bistable reactions (independent of the other space-time variable) for
which we can prove non-existence of transition fronts. These are the first such
results for periodic media which are non-degenerate in a natural sense, and the
spatially periodic examples also prove a conjecture from \cite{DHZ}.
