The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations, and is a natural extension of an equation studied and used by Schoen and Yau in their proof of the positive mass theorem. In this talk we give an extensive study of the generalized equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up behavior are given, and it is shown that blow-up solutions are not unique.