Information security has become a key concern for communication over public channels thus necessitating the use of cryptography to safeguard communication. In this paper, we present a comprehensive review of the asymmetric key algorithms, beginning from the inception of asymmetric cryptography in 1976 till the present day. This paper provides a description of their encryption and decryption operations, points out their security basis, areas of implementation, their strengths and weaknesses during operation. Finally, the paper pinpoints the existing gap based on conclusions drawn from the review, with particular emphasis on an algorithm most suited for industrial application given the current trends in cryptography towards quantum computing. This paper then narrows down on the earnest need for an algorithm that has no trade-off in encryption and decryption speeds, has low computation overhead and is secure enough to withstand quantum algorithm attacks.