Autonomous control for unmanned aerial vehicles (UAVs) has rapidly raised research interest in the last decade. Widely used in civilian, military, and private areas, UAVs have been implemented in surveillance, search and rescue, and delivery tasks, solving problems where a significant space must be covered and traveled. However, using UAVs for navigation problems with full autonomy in control, planning, localization, and mapping tasks remains an open challenge. This paper presents a systematic literature review on object-goal navigation using autonomous UAVs. Object-goal navigation inherits the same navigation challenges, such as control and navigation, in addition to estimating the target's location. The object-search problem requires to understand what object must be found and where it can be located.In this regard, survey taxonomies were found for the tasks and methods behind navigation and target localization problems using UAVs. This review reveals essential issues related to autonomous navigation task dependencies and gaps in UAV development and framework standardization. Open challenges for autonomous UAV control under object-goal navigation must address the research on finding methods for problems, such as autonomy level and comparison metrics, considering safety, ethics, and legal implications.