Traditional acoustic-absorbing materials, such as fiber materials and foam absorbers, do not perform well in the low-frequency regime. They also have disadvantages such as poor durability, limited pressure and temperature resistance, therefore not suitable for applications in harsh industrial and military environments. Acoustic metamaterials could provide new solutions to the above problems: the sound absorption performance of metamaterials may not rely on their intrinsic material properties, but could be freely controlled and engineered with acoustic sub-wavelength structures. In this work, we propose a wedged slow-wave acoustic absorber, which is capable of slowing down the sound speed and achieving strong sound absorption over ultra-broadband frequencies. Through analytical and numerical studies, we show that anisotropic acoustic metamaterials can be designed to have a strong dispersion effect that renders significant slow-sound and thermal-viscous effects. This enables a sound dissipation mechanism that can help break the limits of traditional acoustic-absorbing materials.