  We review the properties of electron shuttles, i.e. nanoelectromechanical
devices that transport electrons one-by-one by utilizing a combination of
electronic and mechanical degrees of freedom. We focus on the extreme quantum
limit, where the mechanical motion is quantized. We introduce the main
theoretical tools needed for the analysis, e.g. generalized master equations
and Wigner functions, and we outline the methods how the resulting large
numerical problems can be handled. Illustrative results are given for current,
noise, and full counting statistics for a number of model systems. Throughout
the review we focus on the physics behind the various approximations, and some
simple examples are given to illustrate the theoretical concepts. We also
comment on the experimental situation.
