  Many physical systems are described by partial differential equations (PDEs).
Determinism then requires the Cauchy problem to be well-posed. Even when the
Cauchy problem is well-posed for generic Cauchy data, there may exist
characteristic Cauchy data. Characteristics of PDEs play an important role both
in Mathematics and in Physics. I will review the theory of characteristics and
bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e.,
those aspects which are invariant under general changes of coordinates. After a
basically analytic introduction, I will pass to a modern, geometric point of
view, presenting characteristics within the jet space approach to PDEs. In
particular, I will discuss the relationship between characteristics and
singularities of solutions and observe that: "wave-fronts are characteristic
surfaces and propagate along bicharacteristics". This remark may be understood
as a mathematical formulation of the wave/particle duality in optics and/or
quantum mechanics. The content of the paper reflects the three hour minicourse
that I gave at the XXII International Fall Workshop on Geometry and Physics,
September 2-5, 2013, Evora, Portugal.
