This talk provides an introductory overview to DNA sequencing, as well as to the algorithms and architectures used for sequence alignment. The presentation will start with a brief introduction to the DNA sequencing process. Afterwards, a description of the optimal and heuristic algorithms for sequence alignment will be presented, as well as the data structures that usually support them. Special attention will be put on approximate string matching algorithms, due to the considerable speedup that may be obtained by using this type of search. Finally, some tools available for biological sequence comparison and for DNA re-sequencing will be presented, as well as some of the hardware structures used to further speed up the alignment process.