An analytical approach for obtaining the buckling strength of 2D and 3D tessellated beam structures is introduced. Moreover, a comprehensive analytical, numerical and experimental study on the elasto-plastic and bifurcation response of a series of novel cellular structures including hierarchical, chiral and anti-chiral honeycomb (2D) structures with triangular, square and hexagonal unit cells are performed. The elastic æproperties of the cellular lattices are obtained through the energy method, and the plastic collapse strength of the structures are determined through the upper-bound collapse limit. æThe methods lead to closed-form relations for the elasto-plastic response of stress under general 2D state of stress. The instabilities in the structures are examined through the beam deflection theory on a representative volume element of the structure to obtain closed-form relations expressing the critical buckling strength of structure under general 2D state of stress. The findings from the analytical studies are confirmed by the finite element method. The proposed method on the buckling strength of 2D and 3D lattice structures under a general loading condition has a far reaching application in the design of lightweight and multifunctional structures.