Constructive type theories generally treat only total functions; partial functions present serious difficulties. In this paper, a theory of partial objects is given which puts partial functions in a general context. Semantic foundations for the theory are given in terms of a theory of inductive relations. The domain of convergence of a partial function is exactly characterized by a predicate within the theory, allowing for abstract reasoning about termination. Induction principles are given for reasoning about these functions, and comparisons are made to the domain theoretic account of LCF. Finally, an undecidability result is presented to suggest connections to a subset of recursive function theory.