The main goal of this paper is to state and present some of the proofs of the Tychonoff theorem, which states that an arbitrary product of compact topological spaces is compact in the product topology. The first three chapters of this thesis prepare the setting just for its proofs. The first chapter serves as a kind of a dictionary of terms concerning the structure of topological spaces and the direct product of topological spaces. In the second chapter, sequences in general topological space are briefly discussed and then generalized introducing the notions of nets and filters. They, unlike sequences, allow sufficient generality when it comes to convergence in a topological space. The third chapter contains basic facts about the compactness of a topological space and examples of such spaces. In addition, it contains characterization of compactness using nets and filters from the previous chapter. Finally, in the last chapter, Tychonoff theorem is stated and two proofs are presented: the first one using nets and filters, and the second one using collections the with finite intersection property.