The last decade was marked by the advent of the concepts of phase gradient and Huygens' meta-surfaces which simple theoretical models have been leading the research on metasurfaces ever since. Meanwhile, theoretical modelling of non-planar metasurfaces have appeared to be exceptionally challenging where it demands accurate analysis of the metasurface geometry. The present work addresses this challenge with a radically different theoretical approach demonstrating how numerical calculation of Green's function can be employed to design conformal metasurfaces of arbitrary geometries within the same framework and without any accommodation. Against the classical concepts , the new approach permits building a metasurface of meta-atoms with electric-only response and significantly simplifying its design and fabrication. The theory is endorsed experimentally by testing several metasurface prototypes for different beam-forming functionalities: radiation with single or multiple beams in any desired directions.