In the first part we will review recent applications of higher structures, focusing on applications of (∞,1)-categories and (∞,n)-categories. Usually such structures are encoded as nice presheaves over categories of "geometric shapes". In the second part we will explain how to obtain several such categories. In the third, informal, part we will consider more examples and will speculate about possible open problems, about general architecture and philosothy of homotopy coherent mathematics.