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Metasurfaces, the ultrathin media with extraordinary wavefront modulation ability, have shown versatile potential in manipulating waves. However, existing acoustic metasurfaces are limited by their narrow-band frequency-dependent capability, which severely hinders their real-world applications that usually require customized dispersion. To address this bottlenecking challenge, we report ultra-broadband achromatic metasurfaces that are capable of delivering arbitrary and frequency-independent wave properties by bottom-up topology optimization. We successively demonstrate three ultra-broadband functionalities, including acoustic beam steering, focusing and levitation, featuring record-breaking relative bandwidths of 93.3%, 120% and 118.9%, respectively. All metasurface elements show novel asymmetric geometries containing multiple scatters, curved air channels and local cavities. Moreover, we reveal that the inversely designed metasurfaces can support integrated internal resonances, bi-anisotropy and multiple scattering, which collectively form the mechanism underpinning the ultra-broadband customized dispersion. Our study opens new horizons for ultra-broadband high-efficiency achromatic functional devices on demand, with promising extension to the optical and elastic achromatic metamaterials.