  We review how phase-field models contributed to the understanding of various
aspects of crystal nucleation including homogeneous and heterogeneous
processes, and their role in microstructure evolution. We recall results
obtained both by the conventional phase-field approaches that rely on spatially
averaged (coarse grained) order parameters in capturing freezing, and by the
recently developed phase-field crystal models that work on the molecular scale,
while employing time averaged particle densities, and are regarded as simple
dynamical density functional theories of classical particles. Besides simpler
cases of homogeneous and heterogeneous nucleation, phenomena addressed by these
techniques include precursor assisted nucleation, nucleation in eutectic and
phase separating systems, phase selection via competing nucleation processes,
growth front nucleation (a process, in which grains of new orientations form at
the solidification front) yielding crystal sheaves and spherulites, and
transition between the growth controlled cellular and the nucleation dominated
equiaxial solidification morphologies.
