  In this paper, we prove the Shafarevich conjecture for proper hyperbolic
polycurves, which is a higher dimensional analogue of that for proper
hyperbolic curves. First, we study theories of proper hyperbolic polycurves
over regular schemes. For example, we generalize the moduli theory of Kodaira
fibrations due to Jost and Yau. We also show the Neron property of proper
smooth models of proper hyperbolic polycurves over Dedekind schemes under an
assumption on residual characteristics. We then apply these extension theories
to the proof of the Shafarevich conjecture for proper hyperbolic polycurves.

Number Theory