  Conformal Riemann mapping of the unit disk onto a simply-connected domain $W$
is a central object of study in classical Complex Analysis. The first complete
proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive,
and theoretical aspects of computing the Riemann map have been extensively
studied since. Carath{\'e}odory Theory describes the boundary extension of the
Riemann map. In this paper we develop its constructive version with explicit
complexity bounds.
