The stability of coherent structures (equilibria, traveling waves, standing waves etc.) is a central problem in fluid dynamics and astrophysics. First, we review three approaches to study linear stability and instability problems, with applications to solitary water waves, barotropic instability in geophysical fluids and turning point principle for gaseous stars. Then we will discuss some unsettled problems including stability of large water waves; coalescence instability of general Cats' eye flows and Zeldovich-Podurets conjecture for stability of relativistic galaxies.