The classical theory of elasticity assumes small strain and small deformation. Small strain will preclude plastic or inelastic deformation (material nonlinearity), and small deformation will preclude buckling. This chapter considers the nonlinear buckling of beams using bifurcation theory via perturbation analysis. The buckling of a column subject to compression had been considered by Euler. The buckling of an arch appears in a very different manner called snap-through buckling. The primary buckling manifests as a pop-up of the central part, whereas secondary buckling appears as plate wrinkles along the edge of the plates. If there is no damping, the motions will not settle to the buckled states and thus the unbuckled state is considered as nonlinearly stable. Snap-through buckling of shallow or flat arches were studied by Timoshenko in 1935 under distributed load and by Biezeno in 1938 under point load.