Abstract: With a wide range of applications in product design and optical watermarking, computational BxDF display has become an emerging trend in the graphics community. In this paper, we analyze the design space of BxDF displays and show that existing approaches cannot reproduce arbitrary BxDFs. In particular, existing surface-based fabrication techniques are often limited to generating only specific angular frequencies, angle-shift-invariant radiance distributions, and sometimes only symmetric BxDFs. To overcome these limitations, we propose diffractive multilayer BxDF displays. We derive forward and inverse methods to synthesize patterns that are printed on stacked, high-resolution transparencies and reproduce prescribed BxDFs with unprecedented degrees of freedom within the limits of available fabrication techniques.