  We give an overview of the applications of noncommutative geometry to
physics. Our focus is entirely on the conceptual ideas, rather than on the
underlying technicalities. Starting historically from the Heisenberg relations,
we will explain how in general noncommutativity yields a canonical time
evolution, while at the same time allowing for the coexistence of discrete and
continuous variables. The spectral approach to geometry is then explained to
encompass two natural ingredients: the line element and the algebra. The
relation between these two is dictated by so-called higher Heisenberg
relations, from which both spin geometry and non-abelian gauge theory emerges.
Our exposition indicates some of the applications in physics, including
Pati--Salam unification beyond the Standard Model, the criticality of dimension
4, second quantization and entropy.
