The world around us is abundant in shapes and lines which can not be described by classical geometry. Nevertheless, thanks to development and discovery of fractals and fractal geometry, a way for their understanding was found. In this thesis, I am going to introduce you with concept of fractals, their classification and classical fractals such as Cantor’s set, the Sierpi´nski triangle and carpet, the Koch curve and snowflake, Pythagorean tree, Julia and Mandelbrot’s sets, and space-filling curves. With the appearance of fractals, a dimension problem was revealed as the previous knowledge about dimensions was insufficient for their description. Hausdorff’s dimension, self-similarity dimension and box-counting dimension are some of the fractal dimension definitions relying on the Hausdorff’s work in 1919. Self-similarity dimension is used in geometric fractals while the box-counting dimension is most commonly used for irregular shapes. Apart from mathematics they have been applied in many other sciences such as physics, biology and most common is computer graphics that helped fractals to become presented to the world.