Natural, semi-natural, and planted forests are a key asset worldwide, providing a broad range of positive externalities. For sustainable forest planning and management, remote sensing (RS) platforms are rapidly going mainstream. In a framework where scientific production is growing exponentially, a systematic analysis of unmanned aerial vehicle (UAV)-based forestry research papers is of paramount importance to understand trends, overlaps and gaps. The present review is organized into two parts (Part I and Part II). Part II inspects specific technical issues regarding the application of UAV-RS in forestry, together with the pros and cons of different UAV solutions and activities where additional effort is needed, such as the technology transfer. Part I systematically analyzes and discusses general aspects of applying UAV in natural, semi-natural and artificial forestry ecosystems in the recent peer-reviewed literature (2018–mid-2020). The specific goals are threefold: (i) create a carefully selected bibliographic dataset that other researchers can draw on for their scientific works; (ii) analyze general and recent trends in RS forest monitoring (iii) reveal gaps in the general research framework where an additional activity is needed. Through double-step filtering of research items found in the Web of Science search engine, the study gathers and analyzes a comprehensive dataset (226 articles). Papers have been categorized into six main topics, and the relevant information has been subsequently extracted. The strong points emerging from this study concern the wide range of topics in the forestry sector and in particular the retrieval of tree inventory parameters often through Digital Aerial Photogrammetry (DAP), RGB sensors, and machine learning techniques. Nevertheless, challenges still exist regarding the promotion of UAV-RS in specific parts of the world, mostly in the tropical and equatorial forests. Much additional research is required for the full exploitation of hyperspectral sensors and for planning long-term monitoring.