Underwater Acoustic Networks (UANs) are widely used in various applications such as climate change monitoring, pollution control and tracking, tactical surveillance and offshore exploration. However, limited consideration is given to the security of such networks, despite the fact that the unique characteristics of UANs make these networks vulnerable to various malicious attacks. In this paper, we address future aspects of how to improve security in UANs. We start by reviewing and discussing the state-of-the-art security threats for underwater networks along with their existing solutions. We then identify the open research issues and challenges in the design of secure protocols for communication in UANs. We propose innovative approaches based on node cooperation, cross-layering, software-defined cognitive networking and context-aware communication in order to effectively provision new or strengthen existing security frameworks in UANs. By using these approaches, we address the problem of detecting malicious behaviours and rogue nodes in order to address the major security issues in UANs. We also investigate the use of a covert channel based detection mechanism which needs to be considered when monitoring or deploying UANs at sea. We believe that the issues raised and future possible solution approaches proposed in this paper will greatly help the researchers contributing towards fortifying security in an inherently in-secure UAN.