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Hybridization is a potent evolutionary process that can affect the origin, maintenance, and loss of biodiversity. Because of its ecological and evolutionary consequences, an understanding of hybridization is important for basic and applied sciences, including conservation biology and agriculture. Herein, we review and discuss ideas that are relevant to the recognition of hybrids and hybridization. We supplement this discussion with simulations. The ideas we present have a long history, particularly in botany, and clarifying them should have practical consequences for managing hybridization and gene flow in plants. One of our primary goals is to illustrate what we can and cannot infer about hybrids and hybridization from molecular data; in other words, we ask when genetic analyses commonly used to study hybridization might mislead us about the history or nature of gene flow and selection. We focus on patterns of variation when hybridization is recent and populations are polymorphic, which are particularly informative for applied issues, such as contemporary hybridization following recent ecological change. We show that hybridization is not a singular process, but instead a collection of related processes with variable outcomes and consequences. Thus, it will often be inappropriate to generalize about the threats or benefits of hybridization from individual studies, and at minimum, it will be important to avoid categorical thinking about what hybridization and hybrids are. We recommend potential sampling and analytical approaches that should help us confront these complexities of hybridization.