Compressed sensing essentially relies on two tenets: the first is that the object we wish to recover is compressible in the sense that it has a sparse expansion in a set of basis functions; the second is that the measurements we make (the sensing waveforms) must be incoherent with these basis functions. This lecture will introduce key results in the field such as a new kind of sampling theorem which states that one can sample a spectrally sparse signal at a rate close to the information rate---and this without information loss.