The article contains a review of publications devoted to the analysis of stability of model objects, rods, plates and shells made of shape memory alloys (SMA) loaded in the modes of martensitic inelasticity or superelasticity. Particular attention is paid to the phenomenon of buckling caused by martensitic phase and (or) structural transformations in these materials. Experimental data are presented to show that the critical buckling loads of this type can be many times lower than the Euler critical loads of elastic buckling corresponding to the minimum (martensitic) values of elastic modules. Various concepts are discussed (fixed or variable phase composition, fixed or variable load, fixed or variable temperature, adiabatic or isothermal buckling) within which these effects can be described. Formulated uncoupled, once coupled and double-coupled formulation of appropriate stability problems. Analytical solutions of boundary value problems of buckling caused by phase and (or) structural transformations for Schenley column on SMA rods, operating on tension – compression and bending of SMA rod, plates and cylindrical shells are obtained. It is shown that the lowest values of the critical parameters are obtained in solving problems in a once-coupled formulation within the framework of the concept of variable load and the assumption of the isothermal nature of buckling (the concept of a fixed temperature). It is established that for the loss of stability of plates from SMA caused by direct phase transformation under the action of bilateral two-parameter compression, the proposition of the convexity of the stability region on the plane of loads is not valid.