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Moduli of rings and quadrilaterals are frequently applied in geometric function theory; see, e.g., the handbook by Kühnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North–Holland, Amsterdam, 2005]. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new -FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz–Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the -FEM algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds.