A general consensus on the origin of quasi-periodic oscillations observed in accreting X-ray binaries is that they are related to the wave dynamics of the accretion disk. We conduct the first systematic study on the detailed dynamical properties of different discoseismic modes, in particular the effects of magnetic fields. We show through local analysis that even a weak magnetic field can "destroy" the self-trapping zone of inertial-gravity modes. The so-called corrugation modes are also strongly affected when the poloidal field approaches equal-partition. Whereas the basic wave properties of inertial-acoustic modes (p-modes) are not affected qualitatively by disk B fields. These modes become unstable large-scale oscillations when the disk vortensity (vorticity divided by surface density) profile has positive gradient at the corotation (where wave pattern speed matches background flow rotating speed) and they are not qualitatively affected by inner disk boundary as long as it has some reflectivity. With numerical simulation, we probe the nonlinear evolution of global overstable disk p-modes and demonstrate that they are quite robust even after non-linear saturation. We find, however, that disc B field can split the corotation resonance and significantly reduce the growth rates of these modes, thus challenging its viability as a model for observed high frequency quasi-periodic oscillations. We employ similar technique to study the dynamics of other astrophysical flows, such as accreting tori and rotating protoneutron stars. Similar suppress- ing effects from magnetic fields are also found in thin accreting tori and rotating protoneutron stars. We found that magnetic fields tend to suppress the Papaloizou-Pringle instability in relatively thin tori, while it can lead to a new instability in relatively thick tori where Papaloizou-Pringle instability cannot operate if the system is non-magnetized. Differentially rotating neutron stars have long been known to be subject to a global non-axisymmetric instability. We show that a toroidal stellar magnetic field can suppress this instability when the B field is a few x 1016 G or stronger.