In this thesis, we have described the ways in which electronic voting is taking place and how electronic voting systems are used. In the introduction, we have highlighted the advantages pointed out by proponents and the disadvantages pointed out by opponents of electronic voting. After that, we have described how various techniques used in electronic voting have been developing through history, and then how a particular system is used in some countries of the world and in the Republic of Croatia. After that we have demonstrated the use of mathematics and cryptography in electronic voting. The main purpose of cryptography is to ensure secure communication between the two sides, which is achieved so that the text being sent is transformed into another text. Firstly, we have identified which mathematical functions are used to map certain words, that is, text transformation. Then we have described some electronic voting methods that aim to ensure correctness, verifiability, user anonymity and the receipt-freeness. In each of these methods we have described the voting process. In the last chapter, we have described three security protocols that can be used in electronic voting. Each of these protocols describes the way participants exchange their messages and explains how participants are sure to communicate with the right person. Following the ”Public Key Infrastructure” protocol, an example of this protocol used in practice is described.