Phylogenetics is concerned with one of the most important problem in biology: reconstructing the evolutionary history of present-day organisms from molecular data, such as DNA, or morphological characters. Hidden from view, in phylogenetics software packages used by biologists, are algorithms implementing stochastic and combinatorial methods on binary trees, as well as more general network structures. The mathematics involved represent a unique confluence of probability theory, discrete mathematics, stochastic methods, and statistical inference, as well as algebraic methods such as group theory. There are many important theoretical and practical problems that arise, such as statistical identifiability of models, consistency and convergence of methods. These problems can only be solved using a multi-disciplinary approach. Phylomania brings together internationally recognised experts, with the aim of discussing the pressing research problems in phylogenetics.