In this paper we will present the design and expected performance for the Electromagnetic and Small Angle Calorimeters (ECAL, SAC) of the PADME experiment. The design of the calorimeters has been optimized for the detection of the final state γ from the annihilation production (and subsequent “invisible” decay) of a “Dark Photon” produced by a positron beam on a thin, low Z target. Beam tests have been made in 2016 and 2017 at the INFN Frascati National Laboratories Linac Beam Test Facility (BTF) with positron beams of energy 100–400 MeV and results are presented. The PADME experiment will be built at the INFN Frascati National Laboratories by the end of 2017 and will be taking data in 2018 (and possibly also 2019). At the moment the collaboration is composed by the following institutions: INFN Roma and “La Sapienza” University of Roma, INFN Frascati, INFN Lecce and University of Salento, MTA Atomki Debrecen, University of Sofia, Cornell University, U.S. William and Mary College.