In order to perform their biological function, many RNA molecules have to take on specific shapes. They fold reliably into these shapes only if their sequence is carefully selected and maintained during the course of evolution.  In this talk, we study the stability of a secondary structure to the mutation of its sequence, in order to better understand the law of statistics any evolution process has to deal with. Towards this goal, we first study the "background" behavior of random RNA sequences. We find that they can be in three different phases: A denatured phase at very high temperatures, a weak glassy phase with a small number of random low energy structures at low temperatures, and a molten phase in which many different structures coexist at intermediate temperatures. We characterize these different phases numerically and analytically.  Knowledge of the above background behavior will be used to deduce how a designed structure (formed due to intentional sequence bias) is degraded as more and more of sequence elements are mutated.

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