Object detection is a critical field in computer vision focusing on accurately identifying and locating specific objects in images or videos. Traditional methods for object detection rely on large labeled training datasets for each object category, which can be time-consuming and expensive to collect and annotate. To address this issue, researchers have introduced few-shot object detection (FSOD) approaches that merge few-shot learning and object detection principles. These approaches allow models to quickly adapt to new object categories with only a few annotated samples. While traditional FSOD methods have been studied before, this survey paper comprehensively reviews FSOD research with a specific focus on covering different FSOD settings such as standard FSOD, generalized FSOD, incremental FSOD, open-set FSOD, and domain adaptive FSOD. These approaches play a vital role in reducing the reliance on extensive labeled datasets, particularly as the need for efficient machine learning models continues to rise. This survey paper aims to provide a comprehensive understanding of the above-mentioned few-shot settings and explore the methodologies for each FSOD task. It thoroughly compares state-of-the-art methods across different FSOD settings, analyzing them in detail based on their evaluation protocols. Additionally, it offers insights into their applications, challenges, and potential future directions in the evolving field of object detection with limited data.