This paper presents a comprehensive review of dispersive Alfven waves in space and laboratory plasmas. We start with linear properties of Alfven waves and show how the inclusion of ion gyroradius, parallel electron inertia, and finite frequency effects modify the Alfven wave properties. Detailed discussions of inertial and kinetic Alfven waves and their polarizations as well as their relations to drift Alfven waves are presented. Up to date observations of waves and field parameters deduced from the measurements by Freja, Fast, and other spacecraft are summarized. We also present laboratory measurements of dispersive Alfven waves, that are of most interest to auroral physics. Electron acceleration by Alfven waves and possible connections of dispersive Alfven waves with ionospheric-magnetospheric resonator and global field-line resonances are also reviewed. Theoretical efforts are directed on studies of Alfven resonance cones, generation of dispersive Alfven waves, as well their nonlinear interactions with the background plasma and self-interaction. Such topics as the dispersive Alfven wave ponderomotive force, density cavitation, wave modulation/filamentation, and Alfven wave self-focusing are reviewed. The nonlinear dispersive Alfven wave studies also include the formation of vortices and their dynamics as well as chaos in Alfven wave turbulence. Finally, we present a rigorous evaluation of theoretical and experimental investigations and point out applications and future perspectives of auroral Alfven wave physics.