Integrated sensing and communications (ISAC) is an emerging critical technique for the next generation of communication systems. However, due to multiple performance metrics used for communication and sensing, the limited degrees-of-freedom (DoF) in optimizing ISAC systems poses a challenge. Reconfigurable intelligent surfaces (RIS) can introduce new DoF for beamforming in ISAC systems, thereby enhancing the performance of communication and sensing simultaneously. In this paper, we propose two optimization techniques for beamforming in RIS-assisted ISAC systems. The first technique is an alternating optimization (AO) algorithm based on the semidefinite relaxation (SDR) method and a one-dimension iterative (ODI) algorithm, which can maximize the radar mutual information (MI) while imposing constraints on the communication rates. The second technique is an AO algorithm based on the Riemannian gradient (RG) method, which can maximize the weighted ISAC performance metrics. Simulation results verify the effectiveness of the proposed schemes. The AO-SDR-ODI method is shown to achieve better communication and sensing performance, than the AO-RG method, at a higher complexity. It is also shown that the mean-squared-error (MSE) of the estimates of the sensing parameters decreases as the radar MI increases.