The first part of this thesis investigates the direct detection of axions in particle physics. A generalized matrix formalism for describing axion-photon mixing in multi-layer systems to all orders in the axion-photon coupling is developed and applied for studying light shinning through a wall (LSW) experiments with and without dielectric layers. It is found that dielectric layers can be placed into two configurations - a transparent and a resonant one. For the transparent configuration, by tuning the distance between the dielectric layers, the experiment can be made to be more sensitive in specific relatively large axion mass ranges. For the ALPS II setup with dielectric layers it is possible to achieve a sensitivity enhancement for axion masses larger than 10−4 eV. Dielectric layers in the resonant case could be used to replace cavities around the (re)generation regions of existing LSW experiments. Then we turn to open axion haloscopes, which aim to detect axions from the dark matter halo. Two methods for effectively calculating the emitted electromagnetic fields in 3D are presented. Both methods represent a significant improvement, as they are much more computationally efficient than a straight forward approach based on standard three dimensional finite element computations. We consider the upcoming MADMAX and BRASS axion haloscope experiments. For the BRASS haloscope we study how axion velocity effects could shift the emitted electromagnetic radiation pattern, while for MADMAX we investigate diffraction, disk tiling and waveguide surroundings. None of the studied 3D effects would be a show stopper for the MADMAX experiment. The second part of the thesis concerns axion quasiparticles (AQs) in topological magentic insulators (TMIs). By AQs we mean quasiparticles, which have the same interaction with the electromagnetic fields as axions from particle physics. AQs in TMIs have not been detected so far. For a future detection via THz transmission spectroscopy a detailed calculation of the expected signal is needed. We present such a calculation and demonstrate that by fitting the future measurements to our signal calculation important material parameters of the TMI can be determined. AQs in TMIs can also be used in order to detect dark matter axions (DAs) since they can resonantly mix with the AQs and photons in TMIs. We present a detailed signal calculation for a DA search with a TMI layer. The calculation takes into account appropriate interface conditions for the electro-magnetic and axion field as well as material losses. Analytical expressions for the resonance width and peak values are presented. For a DA search TMI materials with a relatively small refractive index are advantageous. TMIs with a thickness of a few mm and a surface area of A = 1 m2 can probe QCD axion models for DA masses between 0.7 meV and 3.5 meV. Magnon and photon losses need to be less than 10−5 meV in order not to reduce the emitted signal significantly.