We present fragment assembly of RNA with full-atom refinement (FARFAR), a Rosetta framework for predicting and designing noncanonical motifs that define RNA tertiary structure. In a test set of thirty-two 6-20-nucleotide motifs, FARFAR recapitulated 50% of the experimental structures at near-atomic accuracy. Sequence redesign calculations recovered native bases at 65% of residues engaged in noncanonical interactions, and we experimentally validated mutations predicted to stabilize a signal recognition particle domain.