We present a first detailed survey on underwater and air-water (A-W) wireless communication networks (WCNs) that mainly focuses on the security challenges and the countermeasures proposed to date. For clarity of exposition, this survey paper is mainly divided into two parts. The first part of the paper focuses on the state-of-the-art underwater and A-W WCNs whereby we outline the benefits and drawbacks of the four promising underwater and A-W candidate technologies: radio frequency (RF), acoustic, optical and magnetic induction (MI), along with their channel characteristics. To this end, we also describe the indirect (relay-aided) and direct mechanisms for the A-W WCNs along with their channel characteristics. This sets the stage for the second part of the paper whereby we provide a thorough comparative discussion of a vast set of works that have reported the security breaches (as well as viable countermeasures) for many diverse configurations of the underwater and A-W WCNs. Specifically, we provide a detailed literature review of the various kinds of active and passive attacks which hamper the confidentiality, integrity, authentication and availability of both underwater and A-W WCNs. Finally, we highlight some research gaps in the open literature and identify security related some open problems for the future work.