Barycentric coordinates are extremely useful for deforming shapes. They lead to smooth deformations that are simple and highly efficient to compute. In this chapter, we focus on the special case of planar shape deformation, where the barycentric map can be interpreted as a complex-valued function. We generalize barycentric coordinates from real to complex-valued functions and introduce the notion of complex barycentric coordinates. Complex coordinates can lead to planar barycentric maps with unique shape preserving properties. We provide a general construction for complex barycentric coordinates (Section 7.2.2) and show how to design custom-made complex coordinates (Section 7.2.3). We derive holomorphic coordinates (Section 7.2.1), relying on the rich theory of complex analysis, and explore their intimate relation to conformal maps (Section 7.4).