Accelerator-based heterogeneous architectures, such as CPU-GPU, CPU-TPU, and CPU-FPGA systems, are widely adopted to support the popular artificial intelligence (AI) algorithms that demand intensive computation. When deployed in real-time applications, such as robotics and autonomous vehicles, these architectures must meet stringent timing constraints. To summarize these achievements, this article presents a comprehensive survey of real-time scheduling techniques for accelerator-based heterogeneous platforms. It highlights key advancements from the past ten years, showcasing how proposed solutions have evolved to address the distinct challenges and requirements of these systems. This survey begins with an overview of the hardware characteristics and common task execution models used in accelerator-based heterogeneous systems. It then categorizes the reviewed works based on soft and hard deadline constraints. For soft real-time approaches, we cover real-time scheduling methods supported by hardware vendors and strategies focusing on timing-critical scheduling, energy efficiency, and thermal-aware scheduling. For hard real-time approaches, we first examine support from processor vendors. We then discuss scheduling techniques that guarantee hard deadlines (with strict response time analysis). After reviewing general soft and hard real-time scheduling methods, we explore application- or scenario-driven real-time scheduling techniques for accelerator-enabled heterogeneous computing platforms. Finally, the article concludes with a discussion of open issues and challenges within this research area.