Sometimes we wish to analyze a set of phylogenetic trees or networks themselves, such as a set of gene trees or the output of a Bayesian inference program. One framework for doing this is to consider the trees or networks as points within a geometric moduli space that somehow encodes the non-Euclidean nature of this data. One of the most well-known of these spaces is the Billera-Holmes-Vogtmann (BHV) treespace, which contains all metric trees with the same n leaves. BHV treespace is CAT(0), or non-positively curved, allowing algorithms to quickly compute distances and Frechet means but also resulting in non-Euclidean behaviour like stickiness. In this talk, we look at current results, advantages and disadvantages, and open problems from this geometric approach to statistically analyzing sets of trees.