Several research teams are actively pursuing cloaking/invisibility concepts. One of the prominent groups is at Duke University and the other is at University of California, Berkeley.
The Berkeley team uses a crafted nano-prism of metamaterial atoms that has a negative refractive index for light of near-infrared wavelengths. The Duke team is also uses metamaterials, but their approach is to use layers of circuits built into fiberglass that allow certain frequencies of electromagnetic energy to pass around object. They were able to demonstrate a prototype in 2006 that worked with a narrow range of microwaves and announced that they had been able to refine their approach and apply it to a wider range of frequencies.
This video allows both teams to explain their approaches. Please note that the Duke video is from 2006 and therefore doesn’t contain any demonstrations of their new discoveries.