The authors consider the complex Monge-Ampère equation det(u-ij) = ψ(z, u,▽u) in bounded strictly pseudoconvex domains Ω, subject to the singular boundary condition u = ∞ on (a)Ω. Under suitable conditions on ψ, the existence, uniqueness and the exact asymptotic behavior of solutions to boundary blow-up problems for the complex Monge-Ampère equations are established.