Genome sequence analysis plays a pivotal role in enabling many medical and scientific advancements in personalized medicine, outbreak tracing, and forensics. However, the analysis of genome sequencing data is currently bottlenecked by the computational power and memory bandwidth limitations of existing systems. In this dissertation, we propose four major works, where we characterize the real-system behavior of the genome sequence analysis pipeline and its associated tools, expose the bottlenecks and tradeoffs, and co-design fast and efficient algorithms along with scalable and energy-efficient customized hardware accelerators for the key bottlenecks to enable faster genome sequence analysis. First, we comprehensively analyze the tools in the genome assembly pipeline for long reads in multiple dimensions, uncovering bottlenecks and tradeoffs that different combinations of tools and different underlying systems lead to. Second, we propose GenASM, an acceleration framework that builds upon bitvector-based approximate string matching to accelerate multiple steps of the genome sequence analysis pipeline. We co-design our highly-parallel, scalable and memory-efficient algorithms with low-power and area-efficient hardware accelerators. Third, we implement an FPGA-based prototype for GenASM, where state-of-the-art 3D-stacked memory offers high memory bandwidth and FPGA resources offer high parallelism. Fourth, we propose SeGraM, the first hardware acceleration framework for sequence-to-graph mapping and alignment. We co-design algorithms and accelerators for memory-efficient minimizer-based seeding and bitvector-based, highly-parallel sequence-to-graph alignment. Overall, we demonstrate that genome sequence analysis can be accelerated by co-designing scalable and energy-efficient customized accelerators along with efficient algorithms for the key steps of genome sequence analysis.