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Very recently, two-dimensional (2D) boron sheets (borophene) with rectangular structures were grown successfully on single crystal Ag(111) substrates (Mannix et al 2015 Science http://dx.doi.org/10.1126/science.aad1080 350 http://dx.doi.org/10.1126/science.aad1080 ). The fabricated boroprene is predicted to have unusual mechanical properties. We performed first-principle calculations to investigate the mechanical properties of the monolayer borophene, including ideal tensile strength and critical strain. It was found that monolayer borophene can withstand stress up to 20.26 N m ^−1 and 12.98 N m ^−1 in a and b directions, respectively. However, its critical strain was found to be small. In the a direction, the critical value is only 8%, which, to the best of our knowledge, is the lowest among all studied 2D materials. Our numerical results show that the tensile strain applied in the b direction enhances the bucking height of borophene resulting in an out-of-plane negative Poisson’s ratio, which makes the boron sheet show superior mechanical flexibility along the b direction. The failure mechanism and phonon instability of monolayer borophene were also explored.