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Self-accelerating beams show considerable captivating phenomena and applications owing to their transverse acceleration, diffraction-free and self-healing properties in free space. Metasurfaces consisting of dielectric or metallic subwavelength structures attract enormous attention to acquire self-accelerating beams, owing to their extraordinary capabilities in the arbitrary control of electromagnetic waves. However, because the self-accelerating beam generator possesses a large phase gradient, traditional discrete metasurfaces suffer from insufficient phase sampling, leading to a low efficiency and narrow spectral band. To overcome this limitation, a versatile platform of catenary-inspired dielectric metasurfaces is proposed to endow arbitrary continuous wavefronts. A high diffraction efficiency approaching 100% is obtained in a wide spectral range from 9 to 13 μm. As a proof-of-concept demonstration, the broadband, high-efficiency and high-quality self-accelerating beam generation is experimentally verified in the infrared band. Furthermore, the chiral response of the proposed metasurfaces enables the spin-controlled beam acceleration. Considering these superior performances, this design methodology may find wide applications in particle manipulation, high-resolution imaging, optical vortex generation, and so forth.