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Abstract: The Complex Monge-Ampere equation has the form MA(phi) = mu, where mu is a given Borel measure on a domain in C^n (resp. complex manifold) X, phi is an unknown plurisubharmonic (resp. quasi-plurisubharmonic) function on X, and MA is the Monge-Ampere operator. The existence, uniqueness and regularity questions have been widely studied in the case where mu is absolutely continuous, but much less is known is known if mu is singular (e.g. if mu charges a point). I will discuss some partial results in the singular case and will also describe some examples.