RNA is a fundamental biomolecule that has many purposes within cells. Due to its single-stranded and flexible nature, RNA naturally folds into complex and dynamic structures. Recent technological and computational advances have produced an explosion of RNA structural data. Many RNA structures clearly have regulatory and functional properties. Studying the structure of nascent RNAs is particularly challenging due to their low abundance and long length, but their structures are important because they can influence RNA processing. Precursor RNA processing is a nexus of pathways that determines mature isoform composition and that controls gene expression. In this review, we examine what is known about human nascent RNA structure and the influence of RNA structure on processing of precursor RNAs. These known structures provide examples of how other nascent RNAs may be structured and show how novel RNA structures may influence RNA processing including splicing and polyadenylation. RNA structures can be targeted therapeutically to treat disease.