This thesis seeks to provide an overview on the use of Mellin transforms in Option pricing and to explain related issues. After introducing some basic concepts of Stochastic analysis and Option pricing, we use Mellin transforms as a tool to uncover formulas for pricing of different types of financial derivatives, such as European vanilla and power options, or the American options. Most of this content can be regarded as a summary of existing results on the use of Mellin transforms in option pricing. The main added value of the thesis is the deeper mathematical analysis which most of the preceding studies were lacking. In fact, although Mellin transforms offer an exceedingly convenient tool under the operational (optimistic) approach, the detailed analysis of its use is rather nontrivial.