This chapter deals with the out-of-plane electromagnetic wave propagation in aperiodically-ordered "photonic quasi-crystals" (PQCs). With specific reference on PQC slabs based on the quasi-periodic, 8-fold-symmetric Ammann-Beenker geometry, we review a body of results from our recent studies. More specifically, we study via fullwave numerical simulations the excitation of "guided resonances" in defect-free and point-defected supercells, by looking at the transmittance response for normally-incident plane-wave excitation as well as at the modal structure, and highlighting similarities and differences with respect to the standard periodic case. Finally, we address the fabrication-related issues, and present some preliminary experimental results at infrared wavelengths.Selected Topics in Photonic Crystals and Metamaterials Downloaded from www.worldscientific.com by DEAKIN UNIVERSITY on 08/12/15. For personal use only.