  We construct Hermitian representations of Lie algebroids and associated
unitary representations of Lie groupoids by a geometric quantization procedure.
For this purpose we introduce a new notion of Hamiltonian Lie algebroid
actions. The first step of our procedure consists of the construction of a
prequantization line bundle. Next, we discuss a version of K\"{a}hler
quantization suitable for this setting. We proceed by defining a
Marsden-Weinstein quotient for our setting and prove a ``quantization commutes
with reduction'' theorem. We explain how our geometric quantization procedure
relates to a possible orbit method for Lie groupoids. Our theory encompasses
the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra
actions, actions of families of Lie groups, foliations, as well as some general
constructions from differential geometry.
