The inference of evolutionary histories of species is one of the core problems in biology. The supertree or, more formally, the median tree framework appears to be the most robust approach to the inference of species phylogenies known today. In fact, the majority of popular phylogenomic methods employ the median tree strategy; these include ASTRAL, DupTree, and iGTP. The concept of median trees suggests computing a species phylogeny that minimizes a distance (defined by a fixed phylogenetic measurement) towards a given collection of input trees. The power of the median tree approach and the abundance of different (and often conflicting) evolutionary trees available today, make the phylogenetic comparison measurements a crucial area of research in computational biology. Therefore, a large part of this thesis is dedicated to advancing the algorithmic and theoretical aspects of the phylogenetic comparison measurements. Next, I present our new results on the inference of median trees under the classic phylogenetic comparison measurements. Moreover, I demonstrate novel theoretical guarantees for one of the main median tree methods in the field, ASTRAL. This result ties complex statistical models of evolution to the distance-based species tree inference. Further, I discuss the crucial transition from the tree-paradigm to the network-paradigm that involves the formulation of median network problems. Note that, in contrast to trees, the phylogenetic networks enable modeling of complex evolutionary events, such as horizontal gene transfer, recombination, hybridization, and virus reassortment. I present our most recent advancements in this area that includes a practical study of Influenza viruses.