Due to the manifestation of fascinating physical phenomena and materials science, two-dimensional(2D) materials have recently attracted enormous research interest with respect to the fields of electronics and optoelectronics.There have been in-depth investigations of the nonlinear properties with respect to saturable absorption, and many 2D materials show potential application in optical switches for passive pulsed lasers. However, the Eigen band-gap determines the responding wavelength band and constrains the applications. In this paper, based on band-gap engineering, some different types of 2D broadband saturable absorbers are reviewed in detail, including molybdenum disulfide(MoS_2), vanadium dioxide(VO_2), graphene, and the Bi_2Se_3 topological insulator. The results suggest that the band-gap modification should play important roles in 2D broadband saturable materials and can provide some inspiration for the exploration and design of 2D nanodevices. Due to the manifestation of fascinating physical phenomena and materials science, two-dimensional (2D) materials have recently attracted enormous research interest with respect to the fields of electronics and optoelectronics. There have been in-depth investigations of the nonlinear properties with respect to saturable absorption, and many 2D materials show potential application in optical switches for passive pulsed lasers. However, the Eigen band-gap determines the responding wavelength band and constrains the applications. In this paper, based on band-gap engineering, some different types of 2D broadband saturable absorbers are reviewed in detail, including molybdenum disulfide (MoS_2), vanadium dioxide (VO_2), graphene, and the Bi_2Se_3 topological insulator. The results suggest that the band-gap modification should play important roles in 2D broadband saturable materials and can provide some inspiration for the exploration and design of 2D nanodevices.