Bedford and Taylor showed in 1982 that the complex
Monge-Ampere operator can be well defined (as a regular measure) for locally bounded plurisubharmonic (psh) functions,
and is continuous (in the weak topology) for decreasing
sequences. On the other hand, it is known that this operator
cannot be well defined for all psh functions. We will give a
precise characterization of its domain of definition. It turns
out that in dimension 2 it consists precisely of those psh
functions that belong to the Sobolev space W^{1,2}_{loc}.
We will also discuss a related question on compact K\"ahler
manifolds.