  The Rarita-Schwinger operator is the twisted Dirac operator restricted to
3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are
in addition divergence-free. This is an overdetermined problem and solutions
are rare; it is even more unexpected for there to be large dimensional spaces
of solutions. In this paper we prove the existence of a sequence of compact
manifolds in any given dimension greater than or equal to 4 for which the
dimension of the space of Rarita-Schwinger fields tends to infinity. These
manifolds are either simply connected K\"ahler-Einstein spin with negative
Einstein constant, or products of such spaces with flat tori. Moreover, we
construct Calabi-Yau manifolds of even complex dimension with more linearly
independent Rarita-Schwinger fields than flat tori of the same dimension.
