  Comprehensive investigations of crystalline systems often require methods
bridging atomistic and continuum scales. In this context, coarse-grained
mesoscale approaches are of particular interest as they allow the examination
of large systems and time scales while retaining some microscopic details. The
so-called Phase-Field Crystal (PFC) model conveniently describes crystals at
diffusive time scales through a continuous periodic field which varies on
atomic scales and is related to the atomic number density. To go beyond the
restrictive atomic length scales of the PFC model, a complex amplitude
formulation was first developed by Goldenfeld et al. [Phys. Rev. E 72, 020601
(2005)]. While focusing on length scales larger than the lattice parameter,
this approach can describe crystalline defects, interfaces, and lattice
deformations. It has been used to examine many phenomena including liquid/solid
fronts, grain boundary energies, and strained films. This topical review
focuses on this amplitude expansion of the PFC model and its developments. An
overview of the derivation, connection to the continuum limit, representative
applications, and extensions is presented. A few practical aspects, such as
suitable numerical methods and examples, are illustrated as well. Finally, the
capabilities and bounds of the model, current challenges, and future
perspectives are addressed.
