Spaces becoming noncommutative

Noncommutative geometry has developed over the last three decades building up a large collection of examples and tools which have deep connections with various areas of mathematics and physics. In many situations geometric information which is di cult to encode with traditional tools might be understood by enlarging the class of spaces in consideration in such a way as to allow the presence of \noncommutative coordinates". The noncommutative spaces thus obtained provide a very rich and fruitful framework which often sheds new light on relevant problems and structures.