This chapter reviews some basic aspects pertaining to wave interactions with aperiodically ordered structures, with special emphasis on photonic quasicrystals (PQCs). It provides a presentation, at an introductory level, of the basic properties and applications of PQCs, starting from the geometric aspects, proceeding with a compact review of the topical literature, and the illustration of a series of selected prototype example. The chapter focuses on the more complicated two-dimensional case, intrinsically tied with the theory of “aperiodic tilings” and directly relevant to the examples illustrated. It also reviews some qualitative and semiquantitative aspects necessary to understand the basic aperiodic-order-induced wave phenomena. In order to understand the key properties of aperiodic tilings, one needs to generalize the traditional concepts of order and symmetry, which are intuitively tied to the concept of periodicity. Concerning possible applications to electromagnetics and optical engineering, aperiodic tilings have been proposed as potentially useful geometries in the design of antenna arrays.