New elements of the theory and a preliminary design for a pulse/CW 360 GHz gyroklystron are presented. Issues germane to very high frequency operation of gyroklystron amplifiers, including drive power and limits on cavity Q, are discussed. A specific two cavity design is described. That device had a gain of 33 dB and a beam extraction power of 1 kW with an electronic efficiency of 8% and a power output of 700 W after taking into account Ohmic losses . To produce fast rising pulses, the beam velocity ratio was varied via the modulating anode of a magnetron injection gun. A design for an electron gun through which reduction of the output signal by -50 dB was achieved by a 400 V modulation is given. The input coupler, cavities, electron gun, collector, output coupling optics and collector are described. In addition, issues concerning the fabrication of the high precision, ~2 mm diameter cavities are discussed.