  Characterized by cyclic axisymmetric perturbations to both the magnetic and
fluid parameters, magnetohydrodynamic fast sausage modes (FSMs) have proven
useful for solar coronal seismology given their strong dispersion. This review
starts by summarizing the dispersive properties of the FSMs in the canonical
configuration where the equilibrium quantities are transversely structured in a
step fashion. With this preparation we then review the recent theoretical
studies on coronal FSMs, showing that the canonical dispersion features have
been better understood physically, and further exploited seismologically. In
addition, we show that departures from the canonical equilibrium configuration
have led to qualitatively different dispersion features, thereby substantially
broadening the range of observations that FSMs can be invoked to account for.
We also summarize the advances in forward modeling studies, emphasizing the
intricacies in interpreting observed oscillatory signals in terms of FSMs. All
these advances notwithstanding, we offer a list of aspects that remain to be
better addressed, with the physical connection of coronal FSMs to the
quasi-periodic pulsations in solar flares particularly noteworthy.
