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The full-duplex (FD) communication can achieve higher spectrum efficiency than conventional half-duplex (HD) communication; however, self-interference (SI) is the key hurdle. This paper is the first work to propose the intelligent Omni surface (IOS)-assisted FD multi-input single-output (MISO) FD communication systems to mitigate SI, which solves the frequency-selectivity issue. In particular, two types of IOS are proposed, energy splitting (ES)-IOS and mode switching (MS)-IOS. We aim to maximize data rate and minimize SI power by optimizing the beamforming vectors, amplitudes and phase shifts for the ES-IOS and the mode selection and phase shifts for the MS-IOS. However, the formulated problems are non-convex and challenging to tackle directly. Thus, we design alternative optimization algorithms to solve the problems iteratively. Specifically, the quadratic constraint quadratic programming (QCQP) is employed for the beamforming optimizations, amplitudes and phase shifts optimizations for the ES-IOS and phase shifts optimizations for the MS-IOS. Nevertheless, the binary variables of the MS-IOS render the mode selection optimization intractable, and then we resort to semidefinite relaxation (SDR) and Gaussian randomization procedure to solve it. Simulation results validate the proposed algorithms' efficacy and show the effectiveness of both the IOSs in mitigating SI compared to the case without an IOS.