The concept of flexible tensegrity is discussed in this chapter. Flexible tensegrity structures extend the classical definition of tensegrity to allow the bending and buckling of compression members, resulting in large deformations of these members and off-axial forces. These effects need to be accounted for for highly dynamic applications and large deformations of some tensegrity structures with slender bars. A reduced order model of the bars that accounts for these effects is therefore presented. This model is based on a spring-mass system and allows to capture the elastic buckling behavior of bars in compression accurately and efficiently. Because of its reduced number of degrees of freedom, this model is particularly adapted for large-scale or fast simulations of tensegrity structures, in both dynamic and quasi-static settings. We also discuss the advantage of accounting for the elastic buckling of the bars in the design of tensegrity structures, as a mean to increase their capacity to transfer kinetic energy to elastic energy without structural failure. Some possible applications of flexible tensegrity structures are discussed, including a planetary lander and a 3D tensegrity metamaterial. Both are based on a truncated octahedron tensegrity cell, which has the remarkable property of remaining stable under large deformations despite the buckling of its bars.