Phylogenetic diversity (PD) is a measure of species biodiversity quantified by how much of an evolutionary tree is spanned by a subset of species. In this thesis, we study optimization problems that aim to find species sets with maximum PD in different scenarios, and examine random extinction models under various assumptions to predict the PD of species that will still be present in the future. Optimizing PD with Dependencies is a combinatorial optimization problem in which species form an ecological network. Here, we are interested in selecting species sets of a given size that are ecologically viable and that maximize PD. The NP-hardness of this problem is proved and it is established which special cases of the problem are computationally easy and which are computationally hard. It is also shown that it is NP-complete to decide whether the feasible solution obtained by the greedy algorithm is optimal. We formulate the optimization problem as an integer linear program and find exact solutions to the largest food web currently in the empirical literature. In addition, we give a generalization of PD that can be used for example when we do not know the true evolutionary history. Based on this measure, an optimization problem is formulated. We discuss the complexity and the approximability properties of this problem. In the generalized field of bullets model (g-FOB), species are assumed to become extinct with possibly different probabilities, and extinction events are independent. We show that under this model the distribution of future phylogenetic diversity converges to a normal distribution as the number of species grows. When extinction probabilities are influenced by some binary character on the tree, the state-based field of bullets model (s-FOB) represents a more realistic picture. We compare the expected loss of PD under this model to that under the associated g-FOB model and find that the former is always greater than or equal to the latter. It is natural to further generalize the s-FOB model to allow more than one binary character to affect the extinction probabilities. The expected future PD obtained for the resulting trait-dependent field of bullets model (t-FOB) is compared to that for the associated g-FOB model and our previous result is generalized.