The development of dark energy models has stimulated interest to cosmological
singularities, which differ from the traditional Big Bang and Big Crunch
singularities. We review a broad class of phenomena connected with soft
cosmological singularities in classical and quantum cosmology. We discuss the
classification of singularities from the geometrical point of view and from the
point of view of the behaviour of finite size objects, crossing such
singularities. We discuss in some detail quantum and classical cosmology of
models based on perfect fluids (anti-Chaplygin gas and anti-Chaplygin gas plus
dust), of models based on the Born-Infeld-type fields and of the model of a
scalar field with a potential inversely proportional to the field itself. We
dwell also on the phenomenon of the phantom divide line crossing in the scalar
field models with cusped potentials. Then we discuss the Friedmann equations
modified by quantum corrections to the effective action of the models under
considerations and the influence of such modification on the nature and the
existence of soft singularities. We review also quantum cosmology of models,
where the initial quantum state of the universe is presented by the density
matrix (mixed state). Finally, we discuss the exotic singularities arising in
the brane-world cosmological models.Comment: final version, published in Classical and Quantum Gravity as a
topical revie
