Tensegrity structures are really intriguing: bars floating in the air, without any contact to a solid support, attached only by wires to other bars… that are also floating in the air! The aim of this work is to serve as an introduction to such an atypical kind of structure. It tries to explain everything about the controversial origins and polemic fatherhood; tensegrities from various fields, other than Architecture, structural principles, characteristics, advantages and weakness; precedent and current works and patents; and finally, some new proposals, proving that it is possible to find some applications to architectural and engineering purposes. In conclusion, this work tries to be a guide and reference to a new world of structural possibilities that is blooming and finding its path.