The invention discloses a shift register and a driving method thereof, and a grid driving circuit, and the shift register comprises an input control sub-circuit, an input sub-circuit, a node control sub-circuit, and an output sub-circuit. The input control sub-circuit is respectively connected with the first control end, the second control end, the first enabling end, the first input end and the first node so as to control whether the shift register is started or not; The input sub-circuit is respectively connected with the first node, the first scanning end, the second input end, the second scanning end and the pull-up node; The node control sub-circuit is respectively connected with the pull-up node, the first power supply end, the first reset end, the first clock signal end and the signal output end; And the output sub-circuit is respectively connected with the pull-up node, the second clock signal end and the signal output end. According to the technical scheme provided by the embodiment of the invention, starting and stopping of any stage of shifting register can be realized, and the working stability, the use reliability and the display effect of the display panel are improved.