In this work we consider the role of aperiodic order in different domains of science and technology from an interdisciplinary approach. To start with, we introduce some general classification schemes for aperiodic arrangements of matter. Afterwards, we review the main physical properties and possible applications of quasiperiodic crystals. Several conceptual links between quasiperiodic crystals and the hierarchical structure of biopolymers are then discussed in connection with the charge transfer properties of both biological and synthetic DNA chains. The widespread presence of Fibonacci numbers and the golden mean in different physical contexts is also discussed. Promising technological applications of aperiodic systems are finally reviewed by considering both current and potential applications. In particular, we analyse the capability of exploiting aperiodic order in the design of novel devices based on semiconductor heterostructures and dielectric multilayers.