A general introduction is given to ideal magnetohydrodynamic (MHD) waves, using linear perturbations of ideal MHD equations. Subsequently the shear Alfvén waves are shown to form a discrete spectrum of Global Alfvén Eigenmodes in current-carrying cylindrical plasmas and Toroidal Alfvén Eigenmodes in toroidal plasmas. A comparison between theory and experiment is presented for the observed discrete spectra of Alfvén waves driven by energetic ions in Joint European Torus. The mechanism of excitation of Alfvén instabilities is described by considering particle-to-wave power transfer, and two main mechanisms of TAE-induced re-distribution and losses of energetic ions are discussed.