Among the large variety of complex non-periodic structures, quasicrystals and quasiperiodic distributions play a special role. These structures have some of their physical properties (e.g. dielectric constant, potential, etc.) modulated according to a deterministic non-periodic pattern such as a substitution rules set or a cut & project construction. Such architectures have long been recognized to yield a pronounced long-range order manifesting as an infinite set of crystallographic Bragg peaks and a highly lacunar singular-continuous energy spectrum, with an infinite set of gaps arranged in a multifractal hierarchy. The possibility that such structures also possess distinct topological features has been discussed both in mathematics and physics literature including some descriptions of the spectrum through topological invariants. These topological numbers, emerging from the structural building rules, are known to label the dense set of spectral gaps. We present the topological properties of a finite quasiperiodic chains studied using the scattering and also the diffraction of waves. We show that the topological invariants may be measured from the winding of a chiral scattering phase as a function of a phason structural degree of freedom. Using a Fabry-Perot point of view, this chiral phase is also shown to drive the spectral traverse of conveniently emulated edge states. Furthermore, we present a method to obtain all available topological numbers from the diffraction pattern of a quasicrystal, a method which may be termed topological quasicrystallography. Existing experimental realizations will be addressed, as well as the possible generalizations.