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A series of cyclic shear tests was conducted to reveal the cyclic shear behavior of the interfaces between three types of geomembranes (smooth geomembranes, textured geomembranes, and composite geomembranes) and no-fines concrete. Hysteresis loops and backbone curves were obtained under different vertical pressures and shear-displacement amplitudes. The test results indicate that the textured geomembrane–concrete interface presents higher cyclic friction angles than the other two interfaces. The vertical pressure-dependent shear stiffness and damping ratio were used to evaluate the cyclic shear strength and damping behavior of the different geomembrane–concrete interfaces. In addition, the influence of shear rate and cycle number on the cyclic shear behavior of the geomembrane–concrete interfaces was considered, but there is little difference among experimental data for a specific interface. By analyzing the damage observed for the different geomembranes in the test, the shear behavior of these three interfaces between geomembranes and no-fines concrete was further evaluated. Finally, a reasonable choice of geomembrane is suggested for the stability and integrity of geomembranes in practical geomembrane engineering.