In this article, buckling analysis of doubleorthotropic nanoplates (DONP) embedded in elastic media under biaxial, uniaxial and shear loading is numerically studied. The analysis is based on non-local theory. Both two-variable refined plate theory (TVRPT) and first-order shear deformation plate theory (FSDT) are used to derive the governing equations. Generalized differential quadrature method (GDQM) is utilized to solve the governing equations. In buckling analysis, both in-phase and out-ofphase modes are studied. A graphene sheet is selected as the case study to investigate the numerical results. GDQM results are validated by comparing with the Navier's solutions. After validating the formulation and method of solution, the effect of non-local parameter, geometrical parameters and boundary conditions on the critical buckling load of the double-orthotropic nanoplate are investigated and discussed in detail. It is shown that the effects of non-local parameter for shear buckling are more noticeable than that of biaxial buckling. Moreover, for higher values of non-local parameter, the shear buckling is not dependent on the van der Waals and Winkler moduli.Keywords Two-variable refined plate theory Á Non-local theory Á Buckling analysis Á Generalized differential quadrature method Á Double-orthotropic nanoplate Abbreviations DONP Double-orthotropic nanoplate DQM Differential quadrature method GDQM Generalized differential quadrature method TVRPT Two-variable refined plate theory FSDT First-order shear deformation theory HSDT Higher-order shear deformation theories CPT Classical plate theory NDCBL Non-dimensional critical buckling load