  We establish criteria of stability and instability for the stratified
compressible magnetic Rayleigh--Taylor (RT) problem. More precisely, if under
the stability condition $\Xi <1$, we show the existence of unique solution with
algebraic decay in time for the (compressible) magnetic RT problem with proper
initial data in Lagrangian coordinates. The stability result presents that
sufficiently large vertical (base) magnetic field can inhibit the development
of RT instability. On the other hand, if $\Xi >1$, there exists an unstable
solution to the magnetic RT problem in the Hadamard sense. This shows that the
RT instability still occurs when the strength of base magnetic field is small
or the base magnetic field is horizontal with proper large horizontal period
cell. Moreover, by analyzing the stability condition in magnetic RT problem for
vertical magnetic fields, we can observe that the compressibility destroys the
stabilizing effect of magnetic fields in the vertical direction. Fortunately,
the instability in vertical direction can be inhibited by the stabilizing
effect of pressure, which also plays an important role in the mathematical
proof for stability of the magnetic RT problem. In addition, we will extend the
results in magnetic RT problem to the (compressible) viscoelastic RT problem,
and find that the stabilizing effect of elasticity is stronger than the one of
magnetic fields.
