The NA62 experiment at the CERN SPS aims to measure the branching ratio of the very rare kaon decay $K^+ \rightarrow \pi^+ \nu \bar{\nu}$, collecting $\sim 100$ events with a 10\% background to make a stringent test of the Standard Model. The calorimeter level 0 trigger is used to suppress one of the main backgrounds, the $K^+ \rightarrow \pi^+ \pi^0$ decay, and to select events with a $\pi^+$ in the final state. The calorimeter level 0 trigger identifies clusters in the electromagnetic and hadronic calorimeters. It prepares time-ordered lists of reconstructed clusters together with the arrival time, position, and energy measurements of each cluster. It also provides trigger decisions based on complex combinations of energy and cluster multiplicity. The main parameters of the trigger processor are the high design hit rate (30 MHz) and the required single cluster time resolution (1.5 ns). The calorimeter trigger processor is a parallel system composed of 37 boards, 111 mezzanines and 221 high-performance programmable devices housed in three 9U crates. The design, operation and performances of the calorimeter level 0 trigger are presented.