The secret to invisibility is metamaterials. Traditional substances interfere with visible light waves, and you can't interrupt the flow of visible light waves and remain invisible. So when Harry Potter puts on the invisibility cloak in the first book of the series, he's doing something scientifically sensible—covering himself in a device or substance that prevents his body from interfering with light waves.
Andrea Alú explains how metamaterials are currently being used to practice invisibility on various spectrums, starting with radio waves:
One of Alú's central points is that in order to make a cloak or sheath that doesn't interfere with light waves, you need to cover it in three-dimensional metamaterials that possess a negative refractive index and other design properties suited to the task. And, as you'll note from the tiny sample he showed off in the video, building this material manually isn't practical. What this would mean, in practice, is that we could be decades away from testing invisibility materials.
Debashis Chanda, a nanotechnologist at the University of Central Florida, has found a workaround: if you create a three-dimensional metamaterial "stamp," you can in effect "print" it on very large sheets of material. To put it another way: once a suitable very-small-scale invisibility metamaterial has been designed, it will be potentially adaptable to larger objects. We don't have that invisibility metamaterial yet—but when we do, Chanda's technique may mean that a workable invisibility technology will be adapted from it on a very short timetable. Meanwhile, other large-scale metamaterial projects—ranging from Alú's radio-silent antenna coating material to really confusing interior design options—are now within the realm of possibility.