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In this paper, we apply a reconfigurable intelligent surface (RIS) to a two-way (TW) full-duplex (FD) wireless communication system, where a RIS has <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> reflecting elements and operates as a relay station forwarding signals between two FD terminals. To demonstrate the advantages of RIS compared with traditional relay, we also present the system and signal models of the TW-FD amplify-and-forward relay (AFR) system. We mathematically analyze the performance of the TW-FD-RIS system by deriving the exact closed-form expressions of the average outage probability (AOP) and the average symbol error rate (ASER) in the case <inline-formula> <tex-math notation="LaTeX">$N=1$ </tex-math></inline-formula>, and approximate expressions of the AOP and the ASER in the case <inline-formula> <tex-math notation="LaTeX">$N \ge 2$ </tex-math></inline-formula>. Besides, we provide the exact closed-form expression of the AOP and approximate expression of the ASER of the TW-FD-AFR system. Monte-Carlo simulations validate all derived mathematical expressions. Numerical results demonstrate that using the RIS greatly improves the TW-FD communication system&#x2019;s performance compared with using amplify-and-forward (AF) relay. Specifically, even the RIS with one element (<inline-formula> <tex-math notation="LaTeX">$N=1$ </tex-math></inline-formula>), the AOP and ASER of the TW-FD-RIS system are still lower than those of the TW-FD-AFR system in the low signal-to-noise ratio (SNR) regime. However, in the high SNR regime, the AOP and ASER of the TW-FD-RIS system are higher. In the case <inline-formula> <tex-math notation="LaTeX">$N \ge 2$ </tex-math></inline-formula>, the AOP and ASER of the TW-FD-RIS system are significantly reduced. On the other hand, the impact of residual self-interference (SI) due to FD transmission mode on the performance of TW-FD-RIS and TW-FD-AFR systems is remarkable, making the AOP and ASER of these two systems reach the error floor in the high SNR regime. Fortunately, the usage of RIS with <inline-formula> <tex-math notation="LaTeX">$N \ge 2$ </tex-math></inline-formula> can greatly reduce this impact and avoid the error floor for the TW-FD-RIS system.