The main purpose of this paper is to present ElGamal’s signature scheme and some of its variants. The work is divided into seven chapters, including the introductory part. After we get a brief insight into what the digital signature is and what its application is in our everyday life, for a better understanding of the content of the work the second chapter will cover the basic concepts from the theory of numbers and cryptography. The third and fourth chapters are based on more serious issues that can violate the security of the digital signature and the ways in which it can be prevented. The fifth chapter represents a mathematical problem that occurs in many areas, such as modulo p, which is the problem of discrete logarithm. Through the sixth chapter, we will first get acquainted with ElGamal’s signature scheme published in 1985, and then consider its security. In order to make the scheme clearer, we will go through several examples. The last chapter deals with the variants of the ElGamal signature scheme, which are the Schnorra signature scheme proposed in 1989 and the Digital Signature Algorithm for the first time suggested in 1991.