Electromagnetic metasurface cloaks provide an alternative paradigm toward rendering arbitrarily shaped scatterers invisible. Most transformation-optics (TO) cloaks intrinsically need wavelength-scale volume/thickness, such that the incoming waves could have enough long paths to interact with structured meta-atoms in the cloak region and consequently restore the wavefront. Other challenges of TO cloaks include the polarization-dependent operation to avoid singular parameters of composite cloaking materials and limitations of canonical geometries, e.g., circular, elliptical, trapezoidal, and triangular shapes. Here, we report for the first time a conformal-skin metasurface carpet cloak, enabling to work under arbitrary states of polarization (SOP) at Poincaré sphere for the incident light and arbitrary conformal platform of the object to be cloaked. By exploiting the foundry three-dimensional (3D) printing techniques to fabricate judiciously designed meta-atoms on the external surface of a conformal object, the spatial distributions of intensity and polarization of its scattered lights can be reconstructed exactly the same as if the scattering wavefront were deflected from a flat ground at any SOP, concealing targets under polarization-scanning detections. Two conformal-skin carpet cloaks working for partial- and full-azimuth plane operation are respectively fabricated on trapezoid and pyramid platforms via 3D printing. Experimental results are in good agreement with numerical simulations and both demonstrate the polarization-insensitive cloaking within a desirable bandwidth. Our approach paves a deterministic and robust step forward to the realization of interfacial, free-form, and full-polarization cloaking for a realistic arbitrary-shape target in real-world applications.