The origin of Noncommutative Geometry is twofold. On the one hand there is a wealth of examples of spaces whose coordinate algebra is no longer commutative but which have obvious geometric meaning. The first examples came from phase space in quantum mechanics but there are many others, such as the leaf spaces of foliations, duals of nonabelian discrete groups, the space of Penrose tilings, the Noncommutative torus which plays a role in M-theory compactification and finally the Adele class space which is a natural geometric space carrying an action of the analogue of the Frobenius for global fields of zero characteristic. On the other hand the stretching of geometric thinking imposed by passing to Noncommutative spaces forces one to rethink about most of our familiar notions. The difficulty is not to add arbitrarily the adjective quantum behind our familiar geometric language but to develop far reaching extensions of classical concepts. Several of these new developments are described below with emphasis on the surprises from the noncommutative world.
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