The authors investigate the non-linear stability of stationary solutions of incompressible inviscid stratified fluid flows using the technique initiated by Arnold. The conditions for formal stability of equilibrium flows are established and then the convexity arguments are used to determine conditions for their Lyapunov stability. The resulting non- linear stability criteria involve standard quantities, such as Richardson number, but they differ from linearized stability criteria. Specific examples and the associated Hamiltonian structure for the problems are given.