Non-coding RNA molecules fold into precise base pairing patterns to carry out critical roles in genetic regulation and protein synthesis. We show here that coupling systematic mutagenesis with high-throughput SHAPE chemical mapping enables accurate base pair inference of domains from ribosomal RNA, ribozymes, and riboswitches. For a six-RNA benchmark that challenged prior chemical/computational methods, this mutate-and-map strategy gives secondary structures in agreement with crystallographic data (2 % error rates), including a blind test on a double-glycine riboswitch. Through modeling of partially ordered RNA states, the method enables the first test of an interdomain helix-swap hypothesis for ligand-binding cooperativity in a glycine riboswitch. Finally, the mutate-and-map data report on tertiary contacts within non-coding RNAs; coupled with the Rosetta/FARFAR algorithm, these data give nucleotide-resolution three-dimensional models (5.7 {AA} helix RMSD) of an adenine riboswitch. These results highlight the promise of a two-dimensional chemical strategy for inferring the secondary and tertiary structures that underlie non-coding RNA behavior.