Abstract: A general construction of transfinite barycentric coordinates is obtained as a simple and natural generalization of Floater's mean value coordinates [Flo03, JSW05b]. The Gordon-Wixom interpolation scheme [GW74] and transfinite counterparts of discrete harmonic and Wachspress-Warren coordinates are studied as particular cases of that general construction. Motivated by finite element/volume applications, we study capabilities of transfinite barycentric interpolation schemes to approximate harmonic and quasi-harmonic functions. Finally we establish and analyze links between transfinite barycentric coordinates and certain inverse problems of differential and convex geometry.