Using an atomic force microscope operated inside a transmission electron microscope, we have studied the forces involved in buckling and kinking an individual multiwalled carbon nanotube while observing its structure. In particular, we have measured an individual nanotube's asymptotic critical buckling load and critical kinking load. The buckling results are well described by classical elastic theory, while the observed kinking behavior requires a more involved analysis. Repeated buckling measurements on the same nanotube indicate an extremely high degree of elasticity and set a lower bound on the nanotube's yield strength of 1.7GPa , higher than the yield strength of steel. Plastic deformation of the nanotube was eventually observed following kinking.