Recently, the topic of making objects "invisible" for electromagnetic radiation has gained much attention, following new ideas of using engineered electromagnetic materials with unusual properties for this purpose. This lecture provides a comparative review of the recent developments in this field and discusses the potentials of utilizing these ideas for various microwave and antenna applications. The lecture starts from a review of the notions of invisibility and cloaking and their relations to the classical stealth technologies of reducing radar cross section of targets. Next, we make a historical overview of earlier approaches to techniques allowing reduction of scattering from various objects. Recently proposed solutions for cloaking of objects are reviewed and compared, with the emphasis on the fundamental limitations of their performance. This topic is closely linked to the problem of creating of artificial materials with engineered electromagnetic properties. In particular, materials with equal values of relative permittivity and permeability are of interest. The lecture presents our recent developments of such materials based on mixtures of spiral inclusions and their use for cloaking applications. Furthermore, we discuss the use of electrically dense meshes of transmission lines as cloaking devices. It is shown how new cloaking techniques can be used for applications not necessarily related to cloaking of objects, for instance in new microwave lens antennas or in the design of matched absorbing layers.