Quasicrystals have exotic spectra that are challenging to understand and are the basis of several longstanding problems in spectral analysis. There is also significant excitement about utilising these exotic spectra for wave control applications. In particular, the ability to support many large spectral gaps and exhibit some reported robustness properties (possibly with topological origins) has led several groups to work on enlarging the metamaterial design space beyond just periodic geometries, into the realm of quasicrystals. The first part of this talk will focus on our recent efforts to develop efficient methods for predicting the main spectral gaps in a quasiperiodic waveguide. A common approach is to approximate the spectrum of a quasicrystal with a periodic approximation, known as a supercell. For the specific case of one-dimensional waveguides based on generalised Fibonacci tilings, we have proved that supercell approximations give accurate predictions of the main spectral gaps. This analysis is based on characterising the growth of the underlying recursion relation. We refer to these main gaps as “super band gaps” and have analytically proved their existence in a class of one-dimensional wave systems. The second part of the talk will present recent work to develop applications of the exotic spectral properties of quasicrystals to wave energy harvesting. We have shown that the rainbow trapping phenomenon of graded metamaterials can be combined with the fractal spectra of quasiperiodic waveguides to give a metamaterial that performs fractal rainbow trapping. This is achieved through a graded cut-and-project algorithm that yields a projected geometry for which the effective projection angle (and corresponding local band gap structure) is graded along its length, leading to broadband `fractal’ rainbow trapping. We have demonstrated this principle by designing and building an acoustic waveguide.