In the first chapter, we considered the situation when we have an election with two candidates and we want to find a winner in one round of voting. We introduced the concept of a quota system and its connection with the majority rule. We then described May's theorem, which states that majority rule is characterized among all voting systems by its good properties. In the second chapter, we focused on finding a fair voting system when there are at least three candidates. We explored the similarities and differences in comparison to two candidate elections and used theoretical examples to address the issue of candidates not obtaining more than half of the votes, which potentially leads to a majority of voters rejecting the candidate that has the most support. This led to the consideration of the second, third and subsequent preferences of individual voters and introduced a more complex mathematical model where votes are order relations and social preferences are preference relations.