The Gamma Ray Energy Tracking In-Beam Nuclear Array (GRETINA), capable of determining the energy and position (within 2 mm) of each gamma-ray interaction point and tracking multiple gamma-ray interactions, has been designed. GRETINA will be composed of seven detector modules, each with four highly pure germanium crystals. Each crystal has 36 segments and one central contact instrumented by charge sensitive amplifiers. Two custom designed modules, the Digitizer/Digital Signal Processing (DSP) and the Trigger Timing and Control, compose the electronics of this system. The digitizer/DSP converts the analog information with 14-bit analog to digital converters operating at 100 MS/s, and digitally processes the data to determine the energy and timing information of the gamma interactions with the crystal. Each Digitizer/DSP is controlled by and sends trigger information to the Trigger Timing & Control system through a bidirectional Gbit link. Presently four different trigger algorithms are planned for the trigger system and can be selected for trigger decision. In this paper the details of the electronics and algorithms of the GRETINA data acquisition and trigger system are presented and the performance is reviewed.