Over the last decade, cognitive radios (CRs) have emerged as a technology for improving spectrum efficiency through dynamic spectrum access techniques. More recently, as research interest is shifting beyond 5G communications, new technologies such as reconfigurable intelligent surfaces (RISs) have emerged as enablers of smart radio environments, to further improve signal coverage and spectrum management capabilities. Based on the promise of CRs and RISs, this paper seeks to investigate the concept of adopting both concepts within a network as a means of maximizing the potential benefits available. The paper considers two separate models of RIS-based networks and analyzes several performance metrics associated with the CR secondary user. Monte Carlo simulations are presented to validate the derived expressions. The results indicate the effects of key parameters of the system and the clear improvement of the CR network, in the presence of a RIS-enhanced primary network.