Advancements in Next-Generation Sequencing (NGS) have significantly reduced the cost of generating DNA sequence data and increased the speed of data production. However, such high-throughput data production has increased the need for efficient data analysis programs. One of the most computationally demanding steps in analyzing sequencing data is mapping short reads produced by NGS to a reference DNA sequence, such as a human genome. The mapping program BWA-MEM and its newer version BWA-MEM2, optimized for CPUs, are some of the most popular choices for this task. In this study, we discuss the implementation of BWA-MEM on GPUs. This is a challenging task because many algorithms and data structures in BWA-MEM do not execute efficiently on the GPU architecture. This paper identifies major challenges in developing efficient GPU code on all major stages of the BWA-MEM program, including seeding, seed chaining, Smith-Waterman alignment, memory management, and I/O handling. We conduct comparison experiments against BWA-MEM and BWA-MEM2 running on a 64-thread CPU. The results show that our implementation achieved up to 3.2x speedup over BWA-MEM2 and up to 5.8x over BWA-MEM when using an NVIDIA A40. Using an NVIDIA A6000 and an NVIDIA A100, we achieved a wall-time speedup of up to 3.4x/3.8x over BWA-MEM2 and up to 6.1x/6.8x over BWA-MEM, respectively. In stage-wise comparison, the A40/A6000/A100 GPUs respectively achieved up to 3.7/3.8/4x, 2/2.3/2.5x, and 3.1/5/7.9x speedup on the three major stages of BWA-MEM: seeding and seed chaining, Smith-Waterman, and making SAM output. To the best of our knowledge, this is the first study that attempts to implement the entire BWA-MEM program on GPUs.