There are much better answers above than this one, but:
If you believe fiber bundles are important to classical mathematics, then you probably believe fibrations are, and maybe foliations are, as well. If you don't, note that a foliation of a smooth manifold is a partition decomposition of the manifold into integral submanifolds (roughly, solutions to differential equations). You can't get much more classical than this. In his book Noncommutative Geometry Connes tried to make it clear that to understand the leaf space of a foliation, more is needed than the classical quotient construction, groupoids and noncommutative geometry give more information about a patently classical "space". You probably say: So what? There are other ways. Connes tries then to show us that there is a connection between a fundamental von Neumann algebra invariant (the flow of weights) and one of the key invariants for a codimension 1 foliation (the Godbillon-Vey class), which appears in the first chapter on many introductory accounts of foliations. I find it hard to believe that this is coincidental. For me, this warrants closer investigation.
The index theorem for measured foliations discussed above perhaps grew from a seed like the above mentioned connection. (I wonder what we need to do to get Connes to weigh-in over here at MO?)