  There are notions of L^2-Betti numbers for discrete groups (Cheeger-Gromov,
Lueck), for type II_1-factors (recent work of Connes-Shlyakhtenko) and for
countable standard equivalence relations (Gaboriau). Whereas the first two are
algebraically defined using Lueck's dimension theory,Gaboriau's definition of
the latter is inspired by the work of Cheeger and Gromov. In this article we
give a definition of L^2-Betti numbers of discrete measured groupoids that is
based on Lueck's dimension theory, thereby encompassing the cases of
groups,equivalence relations and holonomy groupoids with an invariant measure
for a complete transversal. We show that with our definition, like with
Gaboriau's,the L^2-Betti numbers of a countable group G coincide with the
L^2-Betti numbers of the orbit equivalence relation of a free action of G on a
probability space. This yields a new proof of the fact the L^2-Betti numbers of
groups with orbit equivalent actions coincide.
