Magnons (spin waves, SWs) are elementary spin excitations in magnetically ordered materials. They are the promising quanta for the transmission and processing of information. Magnons can be coupled to the electromagnetic waves utilized for the wireless communication technologies. A wavelength of magnons is orders of magnitude shorter than that of electromagnetic waves at the same frequency. Therefore, wave-based computing with magnons would be promising for the next generation information technology. For the optimized information processing, artificial magnetic media have the potential to offer advanced controls of SWs. A magnonic crystal, consisting of a periodically arranged nanomagnets, provides a tailored magnon band structure. Periodic magnonic grating couplers (MGCs) efficiently convert centimeter-scaled microwaves to sub-100 nm wavelength SWs. Artificial spin ices (ASIs) are expected to offer reprogrammable magnetization configurations. However, the studies on artificial magnetic materials based on the quasicrystalline and aperiodic arrangement are in their infancy. A quasicrystal, a long range ordered material with a lack of translational invariance, exhibits manifold rotational symmetry. The reciprocal vectors associated with the quasicrystal densely fill out all reciprocal space. These exotic properties would be advantageous for the control of SWs. During my PhD study, I explored SW properties in two dimensional (2D) artificial magnetic quasicrystals (AMQs) to understand the effect of aperiodicity on magnetic properties by means of broadband SW spectroscopy, inelastic light scattering spectroscopy, X-ray magnetic circular dichroism and micromagnetic simulation. By nanofabrication, AMQs made of different magnetic materials were created based on Penrose P2, P3 and Ammann tilings. First, we fabricated AMQs based on aperiodically arranged nanoholes etched into ferromagnetic thin films. Angular-dependent SW spectra exhibited tenfold rotational symmetry reflecting the lattice symmetry of the quasicrystalline nanohole arrangement. Intriguingly, worm-like nanochannels, each exhibiting different SW states, were generated in the AMQs. Our findings imply that the nanohole-based AMQ would become a new class of dense-wavelength division multiplexer. AMQs based on low damping Yttrium iron garnet (YIG) allowed for omnidirectional SW emission thanks to the unconventional rotational symmetry of the quasicrystal. Absorption spectra exhibited forbidden frequency gap openings and a corresponding modification of the magnon density of states, indicating the formation of a magnonic band structure. MGCs, prepared from aperiodically arranged ferromagnetic nanopillars on YIG thin films, allowed also for omnidirectional SW emission with a broad range of wave vectors. The constructive/destructive interference of SWs excited by two emitters allowed for a binary 1/0 output operation. ASIs composed of interconnected ferromagnetic nanobars were found to show non-stochastic switching relevant for reconfigurable functionalities. An ASI integrated on a YIG film showed MGC modes and SW channeling with the presence of an external field. The SW spectra at remnant state exhibited reprogrammable characteristics depending on the magnetic field history. Our study on AMQs is important for the fundamental understanding of quasicrystals as well as fabrication of future magnonic devices targeting at information processing by wave-based computing system on the nanoscale.