Phylogenomics has revealed unexpected levels of disagreement among gene trees in almost every clade examined.  While error in tree inference must play some role in this discordance, biological causes of discordance—primarily incomplete lineage sorting and introgression—provide rich information about the history of organisms.  The current “species tree paradigm” for resolving this discordance can reveal the underlying species relationships when incomplete lineage sorting is the only force acting, and sometimes even when there is introgression.  Here I question the goal of this approach in resolving species relationships, as it explicitly excludes the gene tree discordance from downstream analyses.  I explain how forcing all traits to be analyzed on a single, fixed species tree will often result in incorrect inferences about the direction of evolution and the number of times a trait has evolved.  When levels of introgression are high enough, I argue that the species tree should not even be used as a backbone for comparative analyses, as it will lead to incorrect inferences more often than not.  Though species networks have been proposed as one alternative to species trees, I suggest that we move beyond resolved species trees and networks by using gene tree variation directly in our analyses.  This approach promises to provide a more accurate view of organismal and trait evolution.