Genome analysis fundamentally starts with a process known as read mapping, where sequenced fragments of an organism's genome are compared against a reference genome. Read mapping is currently a major bottleneck in the entire genome analysis pipeline, because state-of-the-art genome sequencing technologies are able to sequence a genome much faster than the computational techniques employed to analyze the genome. We describe the ongoing journey in significantly improving the performance of read mapping. We explain state-of-the-art algorithmic methods and hardware-based acceleration approaches. Algorithmic approaches exploit the structure of the genome as well as the structure of the underlying hardware. Hardware-based acceleration approaches exploit specialized microarchitectures or various execution paradigms (e.g., processing inside or near memory). We conclude with the challenges of adopting these hardware-accelerated read mappers.