Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data values. Traditionally, the coordinates are required to be real and positive. We relax this requirement by allowing the barycentric coordinates to be complex numbers which lead to several powerful advantages. These new coordinates are highly effective in generating conformal mappings from a given planar source region to a planar target region. Complex barycentric coordinates can be used for image manipulation, shape deformation, animation, data interpolation, color interpolation and parameterization.