This paper surveys the literature study on Chosen Ciphertext Attack (CCA) and Chosen Message Attack (CMA) on various signcryption schemes. Signcryption is a new paradigm in public key cryptography that simultaneously fulfills both the functions of digital signature and public key encryption in a logically single step and with a cost significantly lower than that required by the traditional “signature then encryption” approach. Signcryption schemes like ID based, Certificate-less and generalized signcryption must provide the information security against CCA and CMA. To acquire CCA security in confidentiality and CMA security in unforgeability, it should be strengthened against attack. The main objective of this paper is to conduct a study on various security models of different signcryption schemes and their security proof under CCA and CMA.