Dual-space iterative algorithms are algorithms that use alternating modifications of the trial scattering density distribution in direct and reciprocal space to find a solution to the phase problem. The heart of the dual-space algorithms is the iteration scheme: the recipe for combining the modifications in both spaces. An equally important aspect is also the precise definition of these modifications. Numerous iteration schemes exist and these can be combined with an equally rich selection of modifications, leading to a wide range of algorithms. The most well known and the most studied, although by far not the only one, is the charge-flipping algorithm. Dual-space iterative algorithms have found applications in many crystallographic problems. The principal applications in various fields are described with sections devoted to routine structure solution, the solution of the structures of incommensurately modulated crystals and quasicrystals, and structure solution from powder diffraction data.