Genome sequence analysis has enabled significant advancements in medical and scientific areas such as personalized medicine, outbreak tracing, and the understanding of evolution. To perform genome sequencing, devices extract small random fragments of an organism's DNA sequence (known as reads). The first step of genome sequence analysis is a computational process known as read mapping. In read mapping, each fragment is matched to its potential location in the reference genome with the goal of identifying the original location of each read in the genome. Unfortunately, rapid genome sequencing is currently bottlenecked by the computational power and memory bandwidth limitations of existing systems, as many of the steps in genome sequence analysis must process a large amount of data. A major contributor to this bottleneck is approximate string matching (ASM), which is used at multiple points during the mapping process. ASM enables read mapping to account for sequencing errors and genetic variations in the reads. We propose GenASM, the first ASM acceleration framework for genome sequence analysis. GenASM performs bitvectorbased ASM, which can efficiently accelerate multiple steps of genome sequence analysis. We modify the underlying ASM algorithm (Bitap) to significantly increase its parallelism and reduce its memory footprint. Using this modified algorithm, we design the first hardware accelerator for Bitap. Our hardware accelerator consists of specialized systolic-array-based compute units and on-chip SRAMs that are designed to match the rate of computation with memory capacity and bandwidth, resulting in an efficient design whose performance scales linearly as we increase the number of compute units working in parallel. We demonstrate that GenASM provides significant performance and power benefits for three different use cases in genome sequence analysis. First, GenASM accelerates read alignment for both long reads and short reads. For long reads, GenASM outperforms state-of-the-art software and hardware accelerators by 116× and 3.9×, respectively, while reducing power consumption by 37× and 2.7×. For short reads, GenASM outperforms state-of-the-art software and hardware accelerators by 111× and 1.9×. Second, GenASM accelerates pre-alignment filtering for short reads, with 3.7× the performance of a state-of-the-art pre-alignment filter, while reducing power consumption by 1.7× and significantly improving the filtering accuracy. Third, GenASM accelerates edit distance calculation, with 22-12501× and 9.3-400× speedups over the state-of-the-art software library and FPGA-based accelerator, respectively, while reducing power consumption by 548-582× and 67×. We conclude that GenASM is a flexible, high-performance, and low-power framework, and we briefly discuss four other use cases that can benefit from GenASM.