Currently, a significant effort in the world research panorama is focused on finding efficient solutions to a carbon-free energy supply, wave energy being one of the most promising sources of untapped renewable energy. However, wave energy is not currently economic, though control technology has been shown to significantly increase the energy capture capabilities. Usually, the synthesis of a wave energy control strategy requires the adoption of control-oriented models, which are prone to error, particularly arising from unmodelled hydrodynamics, given the complexity of the hydrodynamic interactions between the device and the ocean. In this context, data-driven and data-based control strategies provide a potential solution to some of these issues, using real-time data to gather information about the system dynamics and performance. Thus motivated, this study provides a detailed analysis of different approaches to the exploitation of data in the design of control philosophies for wave energy systems, establishing clear definitions of data-driven and data-based control in this field, together with a classification highlighting the various roles of data in the control synthesis process. In particular, we investigate intrinsic opportunities and limitations behind the use of data in the process of control synthesis, providing a comprehensive review together with critical considerations aimed at directly contributing towards the development of efficient data-driven and data-based control systems for wave energy devices.