This paper studies the implementation of L1 adaptive controller over real-time networks using event-triggering. Event-triggering schedules the data transmission dependent upon errors exceeding certain threshold. We provide event-triggering schemes to characterize the time instants of data transmissions, while using L1 adaptive controller in the feedback loop. Lower bounds on transmission periods are provided. We show that with the proposed event-triggering schemes the states and the input in the networked system can be arbitrarily close to those of a stable reference system by increasing the rate of adaptation and the transmission frequency.