Collective spin excitations propagating in magnetically ordered materials are called spin waves (SWs) or magnons. They are promising for low-power and beyond-CMOS information processing, which do not suffer from the ohmic losses. SWs in ferromagnets (antiferromagnets) cover the few GHz (0.1 to few THz) frequency regime and possess wavelengths in the range of few tens of nanometers which is about five (four) orders of magnitude shorter than electromagnetic waves of the same frequency. This property makes SWs an ideal candidate for applications in microwave and THz technology, essential for on-chip processing of wireless telecommunication signals. Of particular interest are the SWs in antiferromagnets (AFMs) because of the following properties. First, they offer high group velocities due to negligible net magnetization and dipolar effects. Second, their high frequencies due to strong exchange interaction can be utilized to close the THz gap (overcoming the lack of small-sized solid-state THz sources). In this thesis, we investigate SWs in AFMs regarding their potential for i) microwave and ii) THz magnonic devices. i) Yttrium iron garnet (YIG) has been considered the "fruit fly" of magnonics since 1961 due to its low magnetic damping. However, being a ferrimagnet, dipolar interactions cause its magnon band to be inherently anisotropic leading to a reduction of the SW group velocity by an order of magnitude to below 1 km/s in thin films. Here we report the dispersion relation of the low-frequency SW branch of the natural mineral hematite in its canted AFM phase. We determine the damping coefficient of $1.1\times10^{-5}$ from broadband GHz spectroscopy measurements at room temperature. The SW group velocities amount to a few 10 km/s measured by means of wave vector resolved inelastic Brillouin light scattering (BLS). We estimate the SW decay length at small magnetic field of 30 mT to be 1.1 cm. With such a fast SW speed and low damping, our results promote hematite as an alternative and sustainable basis for AFM magnonic devices based on a stable natural mineral. ii) The advancement of compact sources of THz-frequency electromagnetic signals would pave the way for a myriad of novel applications. AFM spintronics has introduced the AFM spin-Hall nano-oscillator (AFM-SHNO), which generates THz signal via the combination of spin-orbit torque (SOT) induced self-sustained THz spin dynamics, spin pumping and the inverse spin Hall effect. In contrast to previous studies on AFM-SHNOs that only examined uniform excitations in the macrospin approximation, we conducted theoretical and micromagnetic simulations on nano-constriction (NC) based SHNOs with uniaxial anisotropy AFM. Our results demonstrate the excitation of propagating SWs by SOT and predict a minimum threshold current with respect to NC size achievable by current nanotechnology. Additionally, we found that the output power of the THz signal generated by uniaxial AFM-based NC-SHNOs is three orders of magnitude higher than that of biaxial AFM-based uniformly excited SHNOs. Our findings highlight the potential of uniaxial AFMs for THz SHNOs, which were previously considered unsuitable. This research advances the understanding of current-driven SWs in AFM-SHNOs, leading to the optimization of practical devices in terms of material, geometry, and frequency. The propagating SWs offer remote terahertz signal generation and efficient synchronization of SHNOs for high-power applications.