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Systematists study how a group of genes or organisms evolved. These biologists now have set their sights on the Tree of Life challenge: to reconstruct the evolutionary history of all known living organisms. A typical phylogenetic reconstruction starts with biomolecular data, such as DNA sequences for modern organisms, and builds a tree, or phylogeny, for these sequences that represents a hypothesized evolutionary history. Finding the best tree for a data set can be a computationally intensive problem. Phylogenetic software for mapping the Tree of Life will require entirely new approaches in statistical models of evolution, high-performance implementations, and data analysis and visualization. The authors have developed polynomial algorithmic techniques for use in their research addressing phylogeny reconstruction from biomolecular sequences, focusing on the accuracy of reconstructions and the use of simulations.