The Segment Anything Model (SAM) is a foundational model for image segmentation tasks, known for its strong generalization across diverse applications. However, its impressive performance comes with significant computational and resource demands, making it challenging to deploy in resource-limited environments such as edge devices. To address this, a variety of SAM variants have been proposed to enhance efficiency while keeping accuracy. This survey provides the first comprehensive review of these efficient SAM variants. We begin by exploring the motivations driving this research. We then present core techniques used in SAM and model acceleration. This is followed by a detailed exploration of SAM acceleration strategies, categorized by approach, and a discussion of several future research directions. Finally, we offer a unified and extensive evaluation of these methods across various hardware, assessing their efficiency and accuracy on representative benchmarks, and providing a clear comparison of their overall performance. To complement this survey, we summarize the papers and codes related to efficient SAM variants at https://github.com/Image-and-Video-Computing-Group/On-Efficient-Variants-of-Segment-Anything-Model.