In this thesis we study fractal dimensions. In the introduction we explain the concept of fractals and we introduce some well-known fractal sets. In the second chapter we define topological dimension and we calculate topological dimensions of some sets, in order to compare them later with their fractal dimensions. In the third chapter we introduce the concept of the Hausdorff measure and we define the Hausdorff dimension. We further study and prove some important properties of Hausdorff dimension. After that, we define the box dimension, we list equivalent definitions of box dimension and then we prove its properties. At the end of the chapter, we shortly compare Hausdorff and box dimension and we give an example of a set for which they are different. Finally, in the fourth chapter, we calculate fractal dimensions of different sets, starting with classical geometric objects. After that, we calculate Hausdorff and box dimensions of some fractal sets. Finally, we compute the box dimension of a spiral which is not really a fractal set, but its box dimension nevertheless describes its density of accumulation at the origin.