It took Dan Shechtman more than two years to get his discovery of an Al-Mn phase with icosahedral diffraction symmetry and sharp Bragg reflections published. A paradigm shift had to take place before this novel ordering state of matter - seemingly contradicting crystallographic laws - could be accepted. Today, more than 25 years later, the existence of quasicrystals is beyond doubt. However, not everything is settled yet. All the factors governing formation, growth, stability and structure of quasicrystals are still not fully understood, nor is it resolved whether their structures are strictly or only on average quasiperiodic, and it is still an open question why only quasicrystals with 5-, 8-, 10- and 12-fold rotational symmetry have been experimentally observed so far. These points will be addressed in this review article.