  The subject of this paper is stochastic acceleration by plasma turbulence, a
process akin to the original model proposed by Fermi. We review the relative
merits of different acceleration models, in particular the so called first
order Fermi acceleration by shocks and second order Fermi by stochastic
processes, and point out that plasma waves or turbulence play an important role
in all mechanisms of acceleration. Thus, stochastic acceleration by turbulence
is active in most situations. We also show that it is the most efficient
mechanism of acceleration of relatively cool non relativistic thermal
background plasma particles. In addition, it can preferentially accelerate
electrons relative to protons as is needed in many astrophysical radiating
sources, where usually there are no indications of presence of shocks. We also
point out that a hybrid acceleration mechanism consisting of initial
acceleration by turbulence of background particles followed by a second stage
acceleration by a shock has many attractive features. It is demonstrated that
the above scenarios can account for many signatures of the accelerated
electrons, protons and other ions, in particular $^3$He and $^4$He, seen
directly as Solar Energetic Particles and through the radiation they produce in
solar flares.
