The main goal of cryptography is to enable secure communication over a public channel; often a secret shared among the communicating parties is used to achieve this. The process by which these parties agree on such a shared secret is called key agreement. In this thesis, we focus on two-party key agreement protocols in the public-key setting and study the various methods used to establish and validate public keys. We pay particular attention to certificateless key agreement schemes and attempt to formalize a relevant notion of security. To that end, we give a possible extension of the existing extended Canetti-Krawzcyk security model applicable to the certificateless setting. We observe that none of the certificateless protocols we have seen in the literature are secure in this model; it is an open question whether such schemes exist. We analyze several published certificateless key agreement protocols, demonstrating the existence of key compromise impersonation attacks and even a man-in-the-middle attack in one case, contrary to the claims of the authors. We also briefly describe weaknesses exhibited by these protocols in the context of our suggested security model
