The Sierpiński product of graphs was introduced as a generalization of Sierpiński graphs, which is a fractal-like family of graphs. The main building blocks of Sierpiński graphs are complete graphs and each next iteration is built in the fractal-like manner of a complete graph. This idea was recently generalized to a graph product, where instead of initially taking just one graph, we build a fractal-like structure with two arbitrary graphs, say G and H. Intuitively this is done in such a way that the product G⊗H has |G| copies of graph H which are connected among themselves according to the edges in G. So we get a graph with local structure of H, but global structure of G (i.e., if we contract all copies of H to a vertex, we get a copy of graph G). This construction is interesting because it may yield graphs with distinguishing number greater than 2. In the talk I will describe the Sierpiński products and list some of their basic properties, which may be useful to answer the open question(s) about the distinguishing number of Sierpiński products.