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Symmetry is fundamental in the description and simulation of quantum systems. Leveraging symmetries in classical simulations of many-body quantum systems often results in an exponential overhead due to the exponentially growing size of some symmetry groups as the number of particles increases. Quantum computers hold the promise of achieving exponential speedup in simulating quantum many-body systems; however, a general method for utilizing symmetries in quantum simulations has not yet been established. In this work, we present a unified framework for incorporating symmetry groups into the simulation of many-body systems on quantum computers. The core of our approach lies in the development of efficient quantum circuits for symmetry-adapted projection onto irreducible representations of a group or pairs of commuting groups. We provide resource estimations for common groups, including the cyclic and permutation groups. Our algorithms demonstrate the capability to prepare coherent superpositions of symmetry-adapted states and to perform quantum evolution across a wide range of models in condensed matter physics and ab initio electronic structure in quantum chemistry. We execute a symmetry-adapted quantum subroutine for small molecules in first quantization on noisy hardware, and demonstrate the emulation of symmetry-adapted quantum phase estimation for preparing coherent superpositions of quantum states in various irreducible representations. In addition, we present a discussion of major open problems regarding the use of symmetries in digital quantum simulations of many-body systems, paving the way for future systematic investigations into leveraging symmetries for practical quantum advantage. The broad applicability and the efficiency of the proposed symmetry-adapted subroutine holds the promise for exponential speedup in quantum simulation of many-body systems.