Morphological instabilities and surface wrinkling of soft materials such as gels and biological tissues are of growing interest to a number of academic disciplines including soft lithography, metrology, flexible electronics, and biomedical engineering. In this paper, we review some of the recent progresses in experimental and theoretical investigations of instabilities that lead to the emergence and evolution of surface wrinkling, folding and creasing under various geometrical constraints (e.g., thin films, sheets, fibers, particles, tubes, cavities, vesicles and capsules) and loading stimuli (e.g., mechanical forces, growth, atrophy, swelling, shrinkage, van der Waals interactions). Some representative theoretical and numerical approaches aimed at modelling the onset of instabilities as well as the postbuckling evolution involving multiple bifurcations and symmetry-breakings are discussed along with the main characteristics and some possible applications of this rich phenomenon.