Keyword spotting (KWS) plays an essential role in enabling speech-based user interaction on smart devices, and conventional KWS (C-KWS) approaches have concentrated on detecting user-agnostic pre-defined keywords. However, in practice, most user interactions come from target users enrolled in the device which motivates to construct personalized keyword spotting. We design two personalized KWS tasks; (1) Target user Biased KWS (TB-KWS) and (2) Target user Only KWS (TO-KWS). To solve the tasks, we propose personalized keyword spotting through multi-task learning (PK-MTL) that consists of multi-task learning and task-adaptation. First, we introduce applying multi-task learning on keyword spotting and speaker verification to leverage user information to the keyword spotting system. Next, we design task-specific scoring functions to adapt to the personalized KWS tasks thoroughly. We evaluate our framework on conventional and personalized scenarios, and the results show that PK-MTL can dramatically reduce the false alarm rate, especially in various practical scenarios.