We give the first subcubic exact algorithms for the RNA folding and Language edit distance problem improving their long-standing cubic running times. RNA folding is one of the most fundamental problems in computational biology that predicts how the nucleotides in a RNA sequence match to create the secondary structure. Language edit distance generalizes both context-free grammar parsing and string edit distance, and computes the edit distance of an input sequence from a context-free language. The problem was first posed by Aho and Peterson in 1972 to design error-correcting parsers, and since then have found numerous applications in databases, programming languages, bioinformatics, machine learning, signal processing etc. RNA folding is a special case of the maximization version of Language edit distance. Both the problems admit dynamic programming solutions that run in time cubic in string length. For over forty years these run times remained unimproved.