The question of the stability of steady state solutions in geophysical fluid flows is addressed through qualitative analysis and quantitative examples. The inviscid linear stability theory of stratified shear flows and the solution of the stability problem using normal modes and Fourier-Laplace transforms are discussed. Two numerical examples are used to illustrate the relationship of various physical parameters to the stability of the system and to trace the development of the instability of the instability for short, intermediate and long times. The examples are (1) two layer fluid of infinite extent with application to the air-sea interface and (2) a two-layer fluid having a free surface and finite depth with application to a salt wedge estuary. The initial-value problem is solved using a power series expansion for short times, superposition of modes for intermediate times and asymptotic analysis for long times. The asymptotic expansion applicable in non-conservative systems is compared with the approximate solution using ray techniques, which are valid in conservative systems, and analytic continuation of the eigenvalues into the complex wavenumber plane. (Author).