Abstract: I will start with a gentle introduction to the noncommutative approach to geometry, stressing the central role that is played by the spectrum of a Dirac operator. I will describe how vector bundles, symmetry groups and connections can naturally (and algebraically) be described in this context. This gives a general framework for gauge theories (in the mathematical sense) but with immediate applications to physics. I will explain that in this general setup it is possible to capture any noncommutative space in terms of an algebra bundle over a commutative space. In the second part of the talk I will describe classes of noncommutative manifolds where one can push this to the geometric level by identifying vertical and horizontal parts of the Dirac operator. This factorization result is compatible with the product structure in KK-theory, but with the additional geometric feature that one may identify an obstructing term that is directly related to the curvature of the underlying fibration.