The plasma particle velocity distributions observed in the solar wind generally show enhanced (non-Maxwellian) suprathermal tails, decreasing as a power law of the velocity and well described by the family of Kappa distribution functions. The presence of non-thermal populations at different altitudes in space plasmas suggests a universal mechanism for their creation and important consequences concerning plasma fluctuations, the resonant and nonresonant wave - particle acceleration and plasma heating. These effects are well described by the kinetic approaches where no closure requires the distributions to be nearly Maxwellian. This paper summarizes and analyzes the various theories proposed for the Kappa distributions and their valuable applications in coronal and space plasmas.