Radio interferometry probes astrophysical signals through incomplete and noisy Fourier measurements. The optimal reconstruction of these signals is an important topic not only for current astronomical imaging but also that of the next generation of radio telescopes, for many of which image dynamic range is a key driver. The theory of compressed sensing demonstrates that incompletely sampled signals, such as those from an interferometer, may be accurately reconstructed when they are sparse or compressible in some basis. The introduction of an explicit sparsity constraint makes the method extremely versatile as it allows prior information on the signal to be introduced. Compressed sensing has been demonstrated to offer significant improvement over standard algorithms, and the flexibility of the framework and its implications for wide-field imaging are compelling, as is its potential for influencing data acquisition methods and improving data storage and transport.