Rapid developments in the emerging field of stretchable and conformable photonics necessitate analytical expressions for boundary conditions at metasurfaces of arbitrary geometries. Here, we introduce the concept of conformal boundary optics: a design theory that determines the optical response for designer input and output fields at such interfaces.Given any object, we can realise coatings to achieve exotic effects like optical illusions and anomalous diffraction behaviour. This approach is relevant to a broad range of applications from conventional refractive optics to the design of the next-generation of wearable optical components. This concept can be generalized to other fields of research where designer interfaces with nontrivial geometries are encountered.In a bulk medium, a wave (e.g., optical, sound, seismic) accumulates phase gradually and propagates without experiencing abrupt variations. At the boundary with another material, however, the wave can experience large -although physically admissible-discontinuities in its reflected and transmitted fields [1,2] Here, we introduce the concept of conformal boundary optics, an analytical method -based on novel, firstprinciple derivations -that allows us to engineer transmission ( ) and reflection ( ) at will for any interface geometry and any given incident wave ( ). By resolving the boundary conditions between two materials at an interface of arbitrary geometry, this method addresses recent developments in nanophotonics with the general technique of differential geometry and coordinates transformation. Unlike transformation