This book is dedicated to the design of the electromagnetic calorimeter of the Mu2e experiment at Fermilab. Mu2e has the ambitious goal to search for the neutrinoless coherent conversion of the muon to an electron in the field of a nucleus. The observation of this physics process would provide the evidence that the theory, called Standard Model, which describes the interactions among elementary particles is not complete. The Mu2e calorimeter has the fundamental function to detect and measure the energy, time and position of impact, of the electrons and identify the signal of conversion electrons, which is expected to be extremely rare, from the electrons produced in the muon decays. The calorimeter has been designed and will be constructed by a collaboration among the Istituto Nazionale di Fisica Nucleare (INFN), the California Institute of Technology (CalTech) and Fermilab (FNAL). I have worked for five months at INFN in Pisa and visited Fermilab in June 2018. The calorimeter is part of a complex experimental apparatus composed of several independent and complementary detectors, including a straw tracker and a plastic scintillator cosmic-ray veto.