Fractal squares and fractal cubes are the simplest self-similar sets. First examples of fractal cubes and squares, such as Sierpinski gasket, Sierpinski carpet, Menger sponge and Cantor dust are well known since the beginning of XX-th century.

A fractal square of order n with a digit set D ⊂ {0, ..., n−1}^2 is the unique compact set K ⊂ [0, 1]^2, which satisfies the equation K = (K + D)/n . The fractal squares are self-similar sets which satisfy the Open Set Condition. They are generated by a finite number of homotheties, therefore their dimension and measure can be easily computed.

Besides general introduction to the topic, the talk is devoted to the structure of Fractal Square Dendrites. We show that fractal square dendrites satisfy one-point intersection property and prove a Theorem which states that there are only 7 classes of Fractal Square Dendrites, according to the type of their main tree and their self-similar boundary.