Metamaterials are artificial materials made of subwavelength elementary cells that give rise to unexpected wave properties that do not exist naturally. However, these properties are generally achieved due to 3D patterning, which is hardly feasible at short wavelengths in the visible and near-infrared regions targeted by most photonic applications. To overcome this limitation, metasurfaces, which are the 2D counterparts of metamaterials, have emerged as promising platforms that are compatible with planar nanotechnologies and thus mass production, which platforms the properties of a metamaterial into a 2D sheet. In the linear regime, wavefront manipulation for lensing, holography, and polarization control has been achieved recently. Interest in metasurfaces operating in the nonlinear regime has also increased due to the ability of metasurfaces to efficiently convert incident light into harmonic frequencies with unusual polarization properties. However, to date, the nonlinear absorption of metasurfaces has been mostly ignored. Here, we demonstrate that plasmonic metasurfaces behave as saturable absorbers with modulation performances superior to the modulation performance of other 2D materials and exhibit unusual polarimetric nonlinear transfer functions. We quantify the link between saturable absorption, the plasmonic resonances of the unit cell and their distribution in a 2D metasurface, and finally provide a practical implementation by integrating the metasurfaces into a fiber laser cavity operating in pulsed regimes driven by the metasurface properties. As such, this work provides new perspectives on ultrathin nonlinear saturable absorbers for applications where tunable nonlinear transfer functions are needed, such as in ultrafast lasers or neuromorphic circuits.