The Spherical Tokamak (ST) concept has become one of the main avenues in magneticnuclear fusion research since STs successfully demonstrated plasma operationat [Beta] = 2P[mu]0=B2~1. Next step ST machines aiming at achieving burning plasmaconditions in high [Beta] plasmas are being planned, such as the Spherical TokamakPower Plant (STPP) and the Component Testing Facility (CTF). Instabilities offast particle-driven Alfven eigenmodes are often observed in present-day STs. Suchinstabilities, driven by fusion-born alpha particles as well as by fast ions producedwith auxiliary heating schemes, in the next step STs may pose a major problem asthese instabilities may affect confinement and losses of the fast ions.A theory of compressional Alfven eigenmodes (CAE) with frequencies above thedeuterium cyclotron frequency,[omega] > [omega]cD, is developed for plasma parameters of aSTPP, and modes in the ion-ion hybrid frequency range, [omega]cT < [omega] < [omega]cD, are alsoinvestigated in order to assess the potential of diagnosing the deuterium-tritium(D-T) ratio. For the 1-D character of a STPP equilibrium with [Beta]~1 , a `hollowcylinder toroidal plasma model is employed for studying CAEs with arbitrary valuesof the parallel wave-vector k[||] = k[.]B/|B|. The existence of weakly-damped CAEs,free of mode conversion, is shown to be associated with the `well in the magneticfield profile, B = B (R), that can exist at the magnetic axis.A significant part of this thesis focusses on the experimentally observed effectsof resonant wave-particle interaction between Alfven waves and fast particles in theMega Amp Spherical Tokamak (MAST) device at the Culham Laboratory, UK, andin the LArge Plasma Device (LAPD) in the University of California, Los-Angeles,USA. New robust experimental scenarios for exciting CAEs in the MAST spherical tokamak are developed, and interpretation of the observed CAEs in the frequencyrange [omega]cD/3 < [omega] < [omega]cD is given in the context of the 1-D ST model and the Dopplershifted cyclotron resonance. The e ciency of the Doppler resonance between coand counter directed fast ions and left and right hand polarised Alfven waves isfurther assessed experimentally on the LAPD device, with probe ions injected inthe presence of Alfv en waves launched by an external antenna.The developed theory of CAEs is then applied to a calculation of the linear kineticdrive of CAEs in the MAST experiments. A model representation of the fast iondistribution function, produced by neutral beam injection (NBI), is used by fittingto the TRANSP Monte-Carlo NBI modelling results. The main free energy sourcesassociated with temperature anisotropy and bump-on-tail are estimated analytically,and the CAE stability boundary is qualitatively assessed.In order to explain the experimentally observed difference between steady-stateand pulsating Alfvenic modes, the non-linear theory of fast particle driven modesnear marginal stability is extended to include dynamical friction (drag). For thebump-on-tail problem, the drag is shown to always give an explosive amplitudeevolution in contrast to diffusion in velocity space in the vicinity of the wave-particleresonance. This is then extended to the case of experimentally observed NBI-driventoroidal Alfven eigenmodes (TAEs) in the MAST machine. The experimentallyobserved differences between TAEs driven by fast ions produced with ion cyclotronresonance heating (ICRH) and NBI are then interpreted. The problem of dragdominated collisions for modes excited by fusion-born alpha particles in burningplasmas such as a STPP and ITER is underlined.