Two methods for the numerical conformal mapping of domains with m < ∞ separated circular holes to domains with m polygonal holes are presented; bounded and unbounded domains are both considered. The methods are based on extensions of the classical Schwarz- Christo el transformation to nitely connected domains. The rst method uses a truncated in nite product expressed in terms of re ections through circles, and is found to have a computational time which increases geometrically with the number of levels of re ection used. The second method uses the boundary behavior of the map to construct a linear system which gives the coe cients of a Laurent series expansion for the map. The second method has a computational time which is polynomial with the number of terms of the truncated series. Both methods require the solution of a non-linear system of equations which gives the correct parameters for the desired map. The solution to the non-linear system is achieved by a numerical continuation (homotopy) method. An application is given. Maps from the circle domains to the canonical slit domains are also computed using similar techniques.